stochastic approach to mid- and long-term forecasting of...

Aalto University

School of Science

Degree Programme of Engineering Physics and Mathematics

Jussi Hirvonen

Stochastic approach to mid- and long-term forecasting of ERCOT real-timeelectricity price

Master’s ThesisEspoo, April 13, 2016

Supervisor: Prof. Pauliina IlmonenInstructor: M.Sc. Jyrki Leino

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Aalto UniversitySchool of ScienceDegree Programme of Engineering Physics and Mathematics

ABSTRACT OFMASTER’S THESIS

Author: Jussi Hirvonen

Title:Stochastic approach to mid- and long-term forecasting of ERCOT real-time elec-tricity price

Date: April 13, 2016 Pages: 111

Professorship: Statistics Code: Mat-2

Supervisor: Prof. Pauliina Ilmonen

Instructor: M.Sc. Jyrki Leino

The purpose of this work is to build understanding of real-time (RT) price creationin Electric Reliability Council of Texas (ERCOT) and construct a stochasticsimulation methodology to create RT price time series several years to future.Simulation methodology takes into account impact of identified drivers of RTprice.

Forecast electricity prices are needed as inputs, for example, for power plantinvestment decisions. In ERCOT, revenues of power plants consist of sellingelectricity in three markets (day-ahead (DA), RT, and ancillary services (AS)markets). Traditionally, mainly DA and partly AS market are considered ininvestment profitability calculations. Flexible power plants can get revenue alsofrom RT market. They can obtain additional profit by benefiting price differencebetween DA and RT markets. Long term price forecasting is needed to supportinvestment decisions due to long lifetime (15+ years) of power plants.

Fundamental and stochastic models are two main classes of electricity price fore-casting models. There are established methods for DA electricity price forecastingand commercial software can do the task. Short term (up to a month), typicallydeterministic, real-time price forecasts are used by market participants, too. How-ever, such solutions do not exist for long-term (3+ years) RT price forecasting.

In this study, statistical analysis is conducted to identify RT price drivers. Sim-ulation methodology is constructed using bootstrap method with several adjust-ments. It is seen that RT price spikes can be largely explained by availablegeneration capacity exceeding demand and DA price. On most common RT pricelevel forecast error of surplus capacity, change speed of net-load, DA price andprevious behaviour of RT price explain price fluctuations. Future values of thechosen explanatory variables can be simulated by conducting a fundamental DAmarket simulation using a dedicated commercial software and calibrated model.

Functioning of simulation method is validated by comparing simulated RT priceseries to historical RT price. Moreover, two case studies are conducted. It isseen that increasing share of wind capacity in ERCOT market increases RT pricevolatility.

Keywords: Forecasting, bootstrap, electricity price, ERCOT

Language: English 2

Aalto-yliopistoPerustieteiden korkeakouluTeknillisen fysiikan ja matematiikan tutkinto-ohjelma

DIPLOMITYONTIIVISTELMA

Tekija: Jussi Hirvonen

Tyon nimi:Sahkon hinnan ennustaminen stokastisin menetelmin

Paivays: 13. huhtikuuta 2016 Sivumaara: 111

Professuuri: Tilastotiede Koodi: Mat-2

Valvoja: Prof. Pauliina Ilmonen

Ohjaaja: M.Sc. Jyrki Leino

Taman tyon tavoitteena on mallintaa paivan sisaista sahkonhintaa Teksasis-sa sijaitsevalla ERCOT-markkinalla. Lisaksi kehitetaan simulointimenetelma,jolla voidaan luoda hinta-aikasarjoja useaksi vuodeksi tulevaisuuteen. Se-littajamuuttujien vaikutus huomioidaan mallinnuksessa.

Sahkonhinnan ennustaminen on tarkeaa esimerkiksi voimalaitosinvestointien kan-nattavuutta laskettaessa. ERCOT:ssa voimalaitokset myyvat sahkoa kolmellamarkkinalla: seuraavan paivan (DA), paivan sisaisella (RT) seka reservituote-markkinalla (AS). Tavallisesti vain DA-markkinan hinta huomioidaan investoin-tilaskelmissa. Nopeat voimalaitokset voivat kuitenkin hyotya merkittavasti RT-markkinahinnan liikkeista ja erityisesti sen ja DA-markkinahinnan erosta.

Sahkonhinnan ennustamiseksi on olemassa kahdenlaisia menetelmia - fundamen-taalisia ja stokastisia. DA-markkinahinnan ennustamiseen kaytetaan yleisesti tie-tokoneohjelmia, joiden antamat tulokset ovat varsin tarkkoja niin lyhyella kuinpitkallakin ennustehorisontilla. RT-markkinahinnalle ei kuitenkaan ole olemassavastaavia pitkan aikavalin ratkaisuja.

Tassa tyossa tutkitaan tilastollisesti RT-markkinahinnan muodostumis-ta. Loydettyihin selittajamuuttujiin perustuen rakennetaan bootstrap-menetelmaa kayttaen stokastinen ennustemalli. Korkeiden hintapiik-kien nahdaan useimmiten tapahtuvan korkean DA-markkinahinnanja alhaisen vapaan sahkontuotantokapasiteetin aikoina. Tavallisimmal-la hintatasolla selittajamuuttujiksi valitaan DA-markkinahinta, vapaansahkontuotantokapasiteetin ennustevirhe, tuulituotannon ylittavan kulu-tuksen muutosnopeus seka edellinen RT-markkinahinta. Selittajamuuttujienarvot simuloidaan sahkomarkkinamallinnukseen tarkoitetulla tietokoneohjelmallaja Markovin ketjuihin perustuvalla stokastisella menetelmalla.

Simulointimenetelman toimivuus varmistetaan ennustamalla menneisyyden hin-toja ja havaitsemalla ennusteet tarpeeksi samanlaisiksi toteutuneiden kanssa. RT-markkinahintaa simuloidaan kahdessa tulevaisuuden skenaariossa. Tuulituotan-non maaran kasvun nahdaan kasvattavan hintavaihteluita RT-markkinalla.

Asiasanat: Ennustaminen, bootstrap, sahkon hinta

Kieli: Englanti

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Acknowledgements

First and foremost, I would like to thank my instructor Jyrki Leino for hisinspiring leadership style, endless expertise, and insightful comments. It hasbeen simply a pleasure to work with him. I will always remember the trip tothe U.S. and all other fun moments we had with Jyrki.

Second, I thank my supervisor professor Pauliina Ilmonen for helping menavigate through this project. Her advice was priceless in both qualitativeand quantitative challenges I faced during this project.

I thank Emilia, Elina, Sofia, Mikko and all other wonderful people in Sales& Marketing team. This thesis would not exist without Kristian Makela andMatti Rautkivi. I thank Wartsila for opportunity to work on this extremelyinteresting research topic.

I am grateful to my brother Roope and all my friends who have occasion-ally interrupted my study evenings for other, more important, aspects of life.I thank Tuuli a thousand times for her patience and being always happy toget me home even after my late nights at work.

Finally, I thank my parents Mia and Pekka for supporting me in every-thing I have decided to do in my life.

Espoo, April 13, 2016

Jussi Hirvonen

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Contents

Abbreviations and Acronyms 7

Notations 10

1 Introduction 141.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2 Research problem . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Structure of the study . . . . . . . . . . . . . . . . . . . . . . 17

2 Electricity markets 182.1 Characteristics of electricity . . . . . . . . . . . . . . . . . . . 19

2.1.1 Demand . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.2 Generation . . . . . . . . . . . . . . . . . . . . . . . . 192.1.3 Transmission . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Different markets . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 Day-ahead market . . . . . . . . . . . . . . . . . . . . 222.2.2 Real-time market . . . . . . . . . . . . . . . . . . . . . 232.2.3 Ancillary services market . . . . . . . . . . . . . . . . . 23

2.3 Forecasting electricity prices . . . . . . . . . . . . . . . . . . . 242.3.1 Day-ahead price forecasting . . . . . . . . . . . . . . . 252.3.2 Real-time price forecasting . . . . . . . . . . . . . . . . 25

2.4 ERCOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.1 Overwiev . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.2 Different markets of ERCOT . . . . . . . . . . . . . . . 27

2.4.2.1 ERCOT Day-ahead energy market . . . . . . 272.4.2.2 ERCOT Real-time energy market . . . . . . . 272.4.2.3 ERCOT Day-ahead ancillary services market 27

5

3 Statistical analysis of RT price 283.1 Analysis of RT electricity price and its potential drivers . . . 28

3.1.1 DA price . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 RT price . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1.3 Spread between RT and DA price . . . . . . . . . . . . 383.1.4 Demand . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1.5 Wind generation . . . . . . . . . . . . . . . . . . . . . 473.1.6 Net-load . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1.7 Non-intermittent generation . . . . . . . . . . . . . . . 563.1.8 Surplus capacity . . . . . . . . . . . . . . . . . . . . . 573.1.9 Forecast error of surplus capacity . . . . . . . . . . . . 583.1.10 Node prices and hub prices . . . . . . . . . . . . . . . . 61

3.2 Selection of simulation methodology . . . . . . . . . . . . . . . 61

4 Stochastic RT price simulation 644.1 Calculating needed historical inputs . . . . . . . . . . . . . . . 654.2 Creating future time series of explanatory variables . . . . . . 69

4.2.1 Day-ahead market simulation . . . . . . . . . . . . . . 694.2.2 Simulating actual values of explanatory variables through

Markov chain and bootstrap methods . . . . . . . . . . 704.2.2.1 Wind generation . . . . . . . . . . . . . . . . 734.2.2.2 Available non-intermittent generation capacity 734.2.2.3 Demand . . . . . . . . . . . . . . . . . . . . . 74

4.3 Stochastic simulation methodology for future RT price . . . . 744.3.1 Simulating RT prices of future spike periods . . . . . . 754.3.2 Simulating other RT prices than spike periods . . . . . 77

4.4 Validating model functioning by comparing historical and sim-ulated values . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.4.1 Actual values of explanatory variables . . . . . . . . . 794.4.2 RT price . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.5 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.5.1 Base scenario and growth scenario . . . . . . . . . . . . 914.5.2 Comparison of results in different scenarios . . . . . . . 94

5 Conclusions 995.1 Topics for future research . . . . . . . . . . . . . . . . . . . . . 100

6 Dynamic time step categories 105

6

Abbreviations and Acronyms

Ancillary services(AS)

Services other than electricity production provided tomarket operator by electricity generators for grid sta-bility maintaining purpose

Annual peak load The highest load of a yearBase load A constant demand of electricity over a long time pe-

riodBase load plant A power plant that usually does not start and stop

within dayBlack-out A situation in which electricity demand of part of or

whole market cannot be servedCapacity factor Mean generation in a period as percentage of genera-

tion capacityClearing Action performed by market operator to match de-

mand and supply based on bids and offers providedby electricity buyers and sellers

Combined cy-cle gas turbine(CCGT)

A gas turbine power plant with a steam turbine thatdrives an additional generator for improved fuel effi-ciency

Congested line A transmission line transmission capacity of which islimiting electric current in it

Congested node A node that is connected to grid by transmission linesthat are often congested

Congestion A situation in which a current in a transmission lineis limited by its transmission capacity

Current operatingplan (COP)

A plan that generating company provides to informERCOT about its intended power plant availability

Day-ahead (DA)market

Electricity market where sold energy is delivered oneday after market clearing

Day-ahead (DA)price

Price of electricity in day-ahead market

7

8

Dispatching Process which optimises generation needed to serveload and provides running schedule to power plants.Conducted by market operator

Energy-only mar-ket

Electricity market type, in which generators are notpaid for their on-line generation capacity, but only forelectricity they sell

ERCOT Electric Reliability Council of Texas. ISO of Texas.Fundamentalmodel

An electricity price forecasting model that is based onmodelling generation of different power plant types

Gas turbine (GT) A gas turbine power plantGate closure Dead-line in minutes for bids and offers before delivery

of electricity, 5 minutes in ERCOT RT marketGrid All electric components (generation, transmission,

consumers) of a marketGrid stability Capability of power system to deliver electricity to

consumers without large frequency deviations andblack-outs

Hub A region of grid that consists of several nodesHub average price Electricity price that is calculated as simple average

of all ERCOT hub pricesHub price Electricity price that is calculated as average of LMPs

within the hubICE Internal combustion engineIndependent powerproducer (IPP)

A private company that owns generation capacity andsells electricity to consumers

Independent sys-tem operator(ISO)

An organization that acts as market operator and co-ordinates power system operation, e.g. ERCOT

Intermediate load Electricity demand that is present for 10 to 18 hoursper day due to regular daily seasonal pattern in de-mand

Intermittent gener-ation

Electricity generation of wind and solar power plants

Load Total electricity demand of marketLoad reduction AS product in which a voluntary electricity consumer

gets compensation for committing to reduce its con-sumption when needed

Locationalmarginal price(LMP)

Electricity price of a node

9

Market operator An organization that matches load and generation inelectricity market

Net-load The load exceeding intermittent generationNodal market An electricity market type in which a typically large

number of nodes have separate prices, e.g. ERCOTNode A place in grid where consumption or generation is

locatedNon-intermittentgeneration

Electricity generation of all other than wind and solarpower plants

Non-spinningreserve

Ancillary service product that requires power plant tobe ready to start generating electricity in 30 minuteswhen needed

Operational flexi-bility

Ability of a power plant to change its output in shorttime

Peak load The load that exceeds intermediate loadPeaking powerplant (peaker)

A power plant that only runs during peak load hours

Price-adder A mechanism that increases electricity price, whensurplus capacity in the system is low. Introduced inERCOT to incentivize building new generation capac-ity.

Price-setter(marginal powerplant)

The power plant with the highest offer price that isdispatched

Price-taker Power plant that is dispatched, but is not price-setterRamp To change output power of a power plant.Ramping capabil-ity

Ability of a power plant to change its output quickly

Ramping con-straint

Inability to ramp up as fast as would be needed. Canbe a problem either on the level of a single power plantor entire system.

Real-time (RT)market

Intra-day electricity market where sold energy is deliv-ered typically very shortly (e.g. 5 minutes in ERCOT)after market clearing

Real-time (RT)price

Price of electricity in real-time market

Regulation-down Ancillary service product to decrease generation whenneeded

Regulation-up Ancillary service product to increase generation whenneeded

10

Responsive reserve Ancillary service product that requires power plantto be ready to start generating electricity in a fewseconds when needed

Risk premium Amount by which the expected value of risky returnmust exceed risk-free return to make a market playerindifferent between risky and risk-free return

Surplus capacity Available generation capacity exceeding loadThree-part supplyoffer

An offer to sell electricity that specifies (i) start-upoffer, (ii) minimum-energy offer, and (iii) energy offercurve

Transmissioncapacity

Maximum electric current of a transmission line

Utility Typically very large company that generates and dis-tributes electricity

Zonal market Electricity market type in which prices are regional,e.g. Nordpool

Notations

RT Real-time electricity priceDA Day-ahead electricity price∆ = RT −DA Spread between day-ahead and real-time price∆hist. Historical ∆ valueIGA Intermittent generation, actual (MW)IGF Day-ahead forecast of intermittent generation (MW)LA Demand, actual (MW)LA Day-ahead forecast of demand (MW)NICA Available non-intermittent generation capacity, actual

(MW)NICF Day-ahead forecast of available non-intermittent gen-

eration capacity (MW)FCA Available generation capacity exceeding demand, ac-

tual (MW)FCF Day-ahead forecast of available generation capacity

exceeding demand (MW)FCFE = FCA −FCF

Forecast error of available generation capacity exceed-ing demand (MW)

NLA = LA− IGA Net-load, actual(MW)NLC Change of net-load from the previous 5-minute period

(MW)C Dynamic time step categoryCspike Dynamic time step category corresponding to spikenC Number of different dynamic time step categories used

in RT price simulationXspike Binary variable that indicates spike period, 1: spike,

0: non-spike∆C ∈ {0, 1, 2, ...} A variable that indicates which group ∆ of certain

period belongs to

11

12

RTC ∈ {0, 1, 2, ...} A variable that indicates which group RT of certainperiod belongs to

nRTC Number of price classes used in RT price samplingn∆C

Number of delta classes used in RT price samplinglRTC A vector that contains limits of price classesl∆C

A vector that contains limits of delta classesDAlimits,C A vector that contains limits of DA price for deter-

mining dynamic time step category CFCFElimit A limit of FCFE for determining dynamic time step

category CNLClimit A limit of NLC for determining dynamic time step

category Clspike,RT A limit above which all RT prices are spike priceslspike,DA A limit above which all DA prices are spike pricesDAlimit,prob A limit of DA price for conditional probability of spike

period startingDAlimit,prob A limit of DA price for conditional probability of spike

period startingFCAlimit,prob A limit of FCA for conditional probability of spike

period startingGRTc Group of historical RT prices that have occurred when

RT class was RTC , C = 0, 1, 2, ...nG∆c Group of historical ∆ values that have occurred ∆

class was ∆C , C = 0, 1, 2, ...nGL A set that includes lengths of historical spike periodsGRTspike A vector that contains RT prices of historical spike

periods in chronological orderSspike Minimum share of spike prices within a spike periodXstart Binary variable, 1: spike period starts, 0: spike period

does not startLB Block length used in simulation of future spike prices

with block bootstrap methodXF Day-ahead forecast value of explanatory variable XXA Actual value of explanatory variable XXFfut. Future day-ahead forecast value of explanatory vari-

able XXAfut. Future actual value of explanatory variable XXFhist. Historical day-ahead forecast value of explanatory

variable XXAhist. Historical actual value of explanatory variable XIGFfut. Future day-ahead forecast of intermittent generation

13

IGAfut. Future actual value of intermittent generationIGFhist. Historical day-ahead forecast value of intermittent

generationIGAhist. Historical actual value of intermittent generationLFfut. Future day-ahead load forecastLAfut. Future actual loadLFhist. Historical day-ahead load forecastLAhist. Historical actual loadNICFfut. Future day-ahead forecast of available non-

intermittent generation capacityNICAfut. Future actual value of available non-intermittent gen-

eration capacityNICFhist. Historical day-ahead forecast of available non-

intermittent generation capacityNICAhist. Historical actual value of available non-intermittent

generation capacityXFC Class of day-ahead forecast of explanatory variable XnXFC

Number of classes XFC of day-ahead forecast used insimulation of actual value of explanatory variable X

lXFCLimits of classes XFC of day-ahead forecast used insimulation of actual value of explanatory variable X

lXAXFCLimits of conditional quantilesQXA used in simulationof actual values of explanatory variable per each classof day-ahead forecast XFC

QXA Conditional quantile of actual value of explanatoryvariable X

q Number of conditional quantiles QXA used in simula-tion of actual values of explanatory variable

Ia(x) Indicator function

Ia(x) =

{1, if x ≥ a

0, otherwise

Chapter 1

Introduction

Forecasting electricity price in competitive markets is a task studied a lot bystatisticians. However, most scientific work on forecasting electricity pricesuntil today has been about day-ahead (DA) electricity price, i.e. the price ofelectricity delivered one day after the market clearing. Flexible power planttechnology allows benefiting movements of Real-Time (RT) price, i.e. theprice of electricity to be delivered in only a few minutes after market clearing.Thanks to their ability to start and stop flexibly for just a few minuteswithout starting cost or impact on maintenance, those power plants can takeadvantage of all price movements. In this work we study fundamentals ofRT price creation in Electric Reliability Council of Texas (ERCOT). We alsobuild a method to create simulated RT price forecasts for several years. Themethod is based on a statistical method known as bootstrapping with severalmodifications to capture the impact of several explanatory variables.

1.1 Background

Power plant investments are typically long-term investments. A major driverof profitability is price for which generated electricity can be sold. Futureprice movements, of course, are not known with certainty at the time thatinvestment decision is made. Traditionally only DA market and ancillaryservices (AS) market, in which different reserve products are traded, havebeen considered in investment decisions (in personal communication withM.Sc. Jyrki Leino, Senior Power System Analyst, Wartsila, 15 March 2016,Turku). However flexible power plants can achieve profits by selling some orall of their generation in RT market. A flexible power plant can also get addi-tional profit by benefiting price differences between markets. For example, itcan sell electricity in DA market and, instead of generating electricity to fulfil

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CHAPTER 1. INTRODUCTION 15

its commitment, buy the commitment back from RT market and not startat all in RT. This strategy is profitable if RT price is lower than marginalgeneration cost of the plant. Wartsila is one manufacturer of flexible powerplants and has a need to understand the value of power plant flexibility inERCOT. Goal of this study is to find the drivers of electricity price in ER-COT RT market and create simulated RT price time series for several years.Forecast price series will be used to determine profitability of power plantinvestments in ERCOT.

ERCOT market was chosen for the study since electricity prices therehave been volatile in recent years. Price volatility has given opportunitiesfor flexible power plants to make profits by making transactions in differentmarkets as described above. Secondly, there is a need to understand whatimpact some changes, such as growing wind generation capacity, will have onprice volatility in ERCOT. RT price volatility is mainly a result of large shareof residential consumers, the demand of which is often inflexible with respectto the price due to low consumption and limited chances to observe electricityprice. Also, the generation in ERCOT market is more difficult to predict thanin many other markets, because of the large share of renewable generationcapacity. Renewable generation is exposed to weather changes, which leadsoccasionally to unexpected events with impact on RT price. Third reasonbehind volatile RT price is low transmission capacity of the interconnectorsthat enable transferring electricity from or to adjacent markets. Therefore,ERCOT is almost an electric island. Unexpected electricity grid or generationoutages cannot be compensated by co-operating with other markets and maylead to high prices.

1.2 Research problem

The purpose of this thesis is to understand RT electricity price creation inERCOT and build a methodology to create forecast price time series for sev-eral years. Due to significant uncertainty related to future events we do nottry to create one precise RT price prediction. Instead, we simulate hundredsof time series and study the distribution of price among all forecast series. Weput a special emphasis on studying the impact of increasing wind generationcapacity on price volatility. The needed steps are (i) building hypothesesand conducting statistical analysis to identify explanatory variables of RTprice movements, (ii) determining quantitative measures of correlation be-tween RT price and each explanatory variable, (iii) constructing a stochasticprice creation method based on correlations and forecast time series of ex-planatory variables, (iv) creating simulated price scenarios using predictions


of explanatory variables provided by ERCOT, and (v) performing ”what-if” analyses to see what impact e.g. demand and wind generation capacitygrowing slightly faster than expected could be expected to have on futureprices.

1.3 Methods

We conduct different statistical analyses to identify drivers of electricity pricein ERCOT RT market. The methods include univariate (e.g. duration curvesand empirical autocorrelation functions) and bivariate (e.g. scatter plots)analyses. We use price data from years 2011-2015 and a lot of other marketdata published by ERCOT and collected and delivered to us by Genscape..Data selection is based on topic understanding achieved through reviewingliterature and interviewing electricity market experts.

To create future price scenarios, we use bootstrap method that was intro-duced by Bradley Efron [7] and has since been used in many contexts, such asmachine learning [2], volatility prediction in stock market modelling [28], andsex determination [26]. Bootstrap is a simple method to apply when reliablehistory data is available. The principle is easy to understand intuitively eventhough the method has rigorous mathematical foundation. It also does notneed too many assumptions about the unknown underlying distribution thatgenerated the observations. For more information about bootstrapping seee.g. [8] or [5].

We use bootstrap method to generate a large number of scenarios forelectricity price in ERCOT RT market for several years. The historical ob-servations of RT price and spread between DA and RT price are used asbootstrapping populations. The impact of different explanatory variablesdetermined by the statistical analyses is taken into account when construct-ing price forecasts for each future time step. Our approach differs in manyways from fundamental electricity price market modelling methods often usedto create short-term price forecasts. Instead of building one prediction of fu-ture price we create several possible scenarios. This approach enables us tomore robustly determine price distribution than in single point estimationby fundamental modelling and explicitly take into account the uncertaintyrelated to future. A large number of price series can also be used in pricerisk evaluation of power plant investment. Statistical approach was chosenbecause of future uncertainty and complicated nature of RT price creationinvolving many drivers. Some of the drivers may be unknown to us and someof them are virtually impossible to observe. Our forecast series are not accu-rate predictions based on explanatory variable values. But neither would be


any prediction based on any explanatory variables, as there is always uncer-tainty about future in electricity markets. However, the long forecast horizontends to cancel out forecast errors made in estimating prices of individualfuture time-steps. Therefore, the estimate of overall price volatility can beexpected to be far more accurate than estimates of RT prices of individualfuture time steps.

1.4 Structure of the study

We start the study by going briefly through in Chapter 2 the functioning ofmodern electricity markets, where price of electricity is determined throughauctions arranged every few minutes. We define many technical terms thatare needed to understand the remaining parts of the work and specificallytake a look at the structure and most important market processes of ERCOT.Next, we briefly review literature on electricity price forecasting. In Chapter3 we conduct statistical analyses to identify drivers of ERCOT RT price. InChapter 4 we build a stochastic model to create simulated RT price series.We apply the method to two future scenarios and compare simulated RTprice series in the scenarios. We see how increasing share of wind generationcapacity in ERCOT market impacts RT price volatility. Finally, in Chapter 5we present our conclusions from the study and propose some ideas for futureresearch.

Chapter 2

Electricity markets

Many electricity markets have liberalised from the 1990s’ regulated fixed-price contract markets to modern competitive markets where electricity priceis determined in auctions organised for example every five minutes [22]. Elec-tricity sellers and buyers send their offers and bids to a computer system heldby market operator. Then market operator clears the market by matchingdemand and supply and dispatches power plants, i.e. schedules power plantsto run so that generation and demand equal every second.

To incentivize building new capacity, some markets have deployed socalled capacity market. In such markets, generators receive payments foreach megawatt of their on-line generation capacity. Energy-only markets donot include such capacity payments and all revenues for generators comefrom selling energy and AS products.

A zonal market is divided to several zones, each of which has its own pricefor electricity. It is possible that price in one zone is consistently higher thanin an other zone. This may happen as a result of congestion, i.e. limited inter-zonal transmission capacity between the two zones. Price difference mayencourage investors seeking greatest returns on their investments to buildnew generation capacity in the high-price zone. Added generation capacitymay lead to prices in zones to converge. One example of zonal market isNordpool [1]. Sometimes significant congestion can occur also within a zone.In zonal market setting intra-zone congestion does not encourage buildingmore capacity in the region with greatest scarcity. Therefore, nodal marketstructure has been introduced in many markets. In such setting there isindividual price for each node, i.e. connection point for load or generation,of the grid. In this work we study one such market, ERCOT.

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CHAPTER 2. ELECTRICITY MARKETS 19

2.1 Characteristics of electricity

With today’s technology, electricity can not be stored for long times in largequantities in a cost-efficient way [31], [33]. Therefore, consumption and gen-eration must be matched every second. This work is done by market operatorby dispatching power plants as needed to serve the demand and, if adjustingis needed, by deploying regulating reserves. Reserves are generation capaci-ties that are able to respond to required quick output changes. Generatorsreceive monetary compensation for providing this adjustment service to thesystem. For a comprehensive presentation of matching electricity demandand generation, see [22] and [25].

2.1.1 Demand

Electricity is consumed by households, firms and public sector in differentways. Aggregate electricity demand exhibits several seasonal patterns thatdiffer by country [22]. In cold countries, such as Finland, demand in wintertime is generally higher than in summer time because of the greater need forelectricity for lighting and warming houses. In warm regions the demand ishigher in summer due to the greater need for air conditioning. Intra-day andweekly patterns are highly similar in many countries. The demand is lowduring people’s low-activity time, i.e. at night and at weekend. The demandis often highest in the working days’ afternoons when public traffic consumesa lot of energy and appliances such as washing machines and computers areswitched on in many homes. However, there are differences in electricityconsumption patterns between consumer groups. Industrial customers haveless clear seasonal patterns than residential customers. Therefore, the dif-ference between the highest and lowest demand of a day is often greater incountries with lower share of industrial electricity consumption. Demand isoften called load, when discussed from generation and transmission systempoint of view.

2.1.2 Generation

In the traditional electricity markets, there were only a few generating com-panies that sold electricity to all consumers [22]. The number of companiesselling electricity has increased rapidly with market liberalisation. For exam-ple ERCOT has 198 certified competitive retail electric providers [12]. Largepublic companies, utilities, may have a fleet of dozens of power plants. Somefirms, independent power producers (IPP), may operate only a single plant.


Table 2.1: Some characteristics of most common power plant types in ER-COT.

TypeInvestment(USD/kW)

Base-load, intermediate-load,peaker, intermittent

Variable operationalcost (USD/MWh)

Typical plantsize (MW)

Coal 1000 base 25 2000Nuclear 4000 base 7 3000CCGT 700 base/intermediate 50 1000OCGT 500 peaker 70 100Engine 600 intermediate/peaker 60 100Wind 1000 intermittent 5 100Solar 3000 intermittent 6 10

All types of power plants have their own role in the well-functioningpower system. So called base load plants can generate electricity at verylow cost and their output is kept stable regardless of load. Nuclear and coalpower plants, as well as combined cycle gas turbine plants (CCGT) are oftenbase load plants. Base load plants have often very high generation capacityand require high construction investment. Intermediate plants operate onlywhen load exceeds the total capacity of the base load plants. Their costper start and required initial investment are often high. Intermediate plantscan not generate electricity at as low a cost as base load plants. Therefore,the electricity price is generally higher, when intermediate plants are neededto serve the load. For example, CCGTs may be operated as intermediateplants. Peaking power plants run only during high-demand periods. Theyhave lower cost per start than base load plants and intermediate plants andthey are capable of adjusting their output power very fast. Initial investmentis often low compared to base load and intermediate plants, but they havehigher marginal generation cost resulting from poor fuel-efficiency. Suitablepeaker technologies include gas turbines (GT) and internal combustion en-gines (ICE).

Power plants that use renewable energy sources can generate electricityat very low cost since their fuel is free. In many markets their generation isalways used when it is available [24]. However, generation of wind and solarpower plants is intermittent by nature as their electricity production stopsimmediately, when the sun stops shining or there is no more wind. Generationof wind and solar power plants is therefore called intermittent generation. Inmany markets renewable technologies are subsidised by political decision,because they would not be economic choice otherwise. Table 2.1 shows somecharacteristics of different power plant types discussed above [22], [21], [19].Numbers are only directional.

The share of load that can not be served by intermittent plants is callednet-load. It can change in two ways, (i) when load changes or (ii) generationof intermittent plants changes. Net-load changes require ramping from non-


intermittent plants, which requires some degree of operational flexibility fromthem. Current trend of adding intermittent generation in many markets willmake net-load changes faster and larger. Therefore, need for flexible gener-ation capacity will increase in future. Figure 2.1 illustrates load, net-loadand intermittent generation of one example day in ERCOT. Graph showsload increasing from 25 GW to almost 45 GW during the day. Simultaneousdecrease in wind generation from 12 GW to less than 4 GW makes net-loadgrowth even faster than load growth. This kind of rapid net-load growthrequires non-intermittent power plants to ramp up fast.

Figure 2.1: Load, wind generation, and net-load of October 22 2015 in ER-COT. Load is the black line on top of the graph. Wind generation is thegreen region on top of the graph. Net-load is the blue region below windgeneration. Load is the sum of net-load and wind generation. Load increasesuntil it reaches its maximum of approximately 44000 MW. Wind generationis strong in the night, but decreases at the same time that load increasesmaking the net-load change even more rapidly than load. Steep increase ofnet-load is seen as rapid widening of the blue region at night and in themorning.

2.1.3 Transmission

Electricity is transmitted to the place of consumption as electric current viatransmission lines. However, there is upper limit for current called trans-


mission capacity of transmission line. All transmission lines and nodes of amarket together are called grid. A node is a place where load or generationis electrically connected to a transmission line. A transmission line which istransmitting at its full transmission capacity, is called congested line, and thesituation when one or more transmission line of a grid is congested is calledcongestion. A congested node is a node that is connected to a transmissionline that is often congested. It is a task of market operator to ensure thatthe price of electricity for the users in each node is the cheapest possible. Ifa node is connected to a low-cost generation unit by transmission lines withabundant capacity, the price in that node will probably be often cheaperthan price in an other node that is very congested and has only high-costgenerating units near it. The prices of two nodes can differ, if and only if,all transmission lines connecting these nodes are congested or not function-ing at all due to line outages. Therefore, one way to reduce market priceof electricity is reducing congestion by increasing transmission capacity oftransmission lines that are most often congested. It would be technicallypossible to remove all congestion by adding transmission lines with highertransmission capacity, but it might require high investments. As a result thetotal cost of electricity to the users might be even higher than in situationwhere occasional differences in prices of congested nodes occur.

2.2 Different markets

In competitive markets, such as ERCOT, electricity price is determined bydemand and supply, and is free to fluctuate as the two change. Electricityis sold in two main markets: DA market and RT market. In addition thereis AS market, where market operator buys several reserve products fromgenerators. Reserves are needed to ensure that unexpected events do notrisk grid stability, leading at worst case to black-outs.

2.2.1 Day-ahead market

In DA market, a daily auction is arranged, where electricity buyers and sellersprovide their bids. The clearing of market is executed by market operatorone day before actual delivery of electricity. Therefore, by definition, DAmarket commitments are future contracts, where the underlying is certainamount of electricity and delivery time is specified to a certain time of nextday. In many markets, such as ERCOT, DA market is hourly. Hourly DAmarket means that market participants can give separate bids for each hourof the next day [20].


2.2.2 Real-time market

Even the best predictions of electricity demand made for next day are justestimates and many things can change before the delivery time of DA con-tracts. Matching demand and generation exactly for next day would bevirtually impossible and often one or both of the two would differ. Differ-ence between generation and demand might lead occasionally to a situationwhere a part of the regions load would have to be shed to avoid black-outin the whole grid. For that reason there is RT electricity market, which issometimes called intra-day market. In RT market electricity auctions arearranged very often, even every five minutes. Delivery of electricity happense.g. five minutes after gate closure, which is a dead-line for sending bids andoffers. Dispatching is executed by market operator in a similar way to DAmarket. All plants can participate in the market, but slow-ramping plantscan only be dispatched respecting their ramping limits.

A plant that has sold electricity in DA market can sometimes use RTprice movements to its advantage. Flexibility is required to make operativedecisions only a few minutes before action is needed. Often power plantssimply fulfil their DA commitments by generating the electricity that theyhave sold. Alternatively they can buy electricity in RT market and deliver itto the grid (not physically), so fulfilling their commitment without startingat all. This is more profitable strategy, if buying the electricity from RTmarket is cheaper for the power plant than generating it themselves. Suchsituation occurs, when RT price is lower than the marginal generation costof the power plant.

2.2.3 Ancillary services market

Unexpected generator or transmission outages can occur after RT-marketclearing or load may be slightly different than was expected. If there wouldnot be any way to compensate for these changes in supply and demand,there would be a good chance that the generation and load would not equal.It would be seen in the frequency of alternating current, and might leadto black-outs in extreme cases. AS products are used if system stability isin danger. Market operator buys these products from generators as com-mitments to increase or decrease their output when needed. Regulation upcapacity sold by a power plant means that it must increase its output bycertain amount, whenever the frequency of alternating current falls belowa certain threshold. Regulation down is a commitment of a power plant todecrease its output by certain amount whenever the frequency exceeds a cer-tain threshold determined by market operator. Regulation products can be


provided only by power plants that can respond to changes in frequency veryquickly, only in a few seconds. There are also reserves that can be deployedby market operator in a situation where a power plant trips, or for anotherreason generation suddenly drops significantly. One product that can be usedto avoid load shedding is load reduction. It is a commitment of a voluntaryelectricity consumer, e.g. a large industrial site, to reduce its consumptionwhen needed [18]. Some power plants, such as ICEs, can get a large shareof their revenue from selling AS products on top of their energy sales, bye.g. selling their entire output in the DA market and regulation-down in ASmarket. Some power plant types, e.g. nuclear, hardly ever participate inAS market. For a thorough discussion of frequency response, see [21]. Thepurpose and functioning of different reserves are discussed in [25]. In thiswork we do not study AS market or prices.

2.3 Forecasting electricity prices

Ability to forecast electricity prices is important for many parties. Powerplant owners do investment decisions based on long-term forecasts and oper-ative decisions based on short-term forecasts of future prices. For example,building new flexible generation capacity requires enough price volatility tobe profitable. Regulators and politicians try to create market rules so thatthey encourage building sufficient generation capacities of different types sothat load can be served reliably and cost-effectively.

Different methods to create price forecasts exist. Selection of methodol-ogy depends on intended purpose of forecast future price time series. Twomain approaches to price forecasting are deterministic and stochastic models.Deterministic models are used to create one prediction of time series, somekind of maximum likelihood estimate. Deterministic models are often usedfor short-term (up to a month) forecasting. One example of use of deter-ministic price modelling is creation of power plant bidding strategy for DAmarket by a trader working for a utility. Stochastic models are based onsimulating many forecast price time series, and studying the distribution ofsimulated time series. For example, 95 % confidence intervals of electricityprice of next years Christmas could be created using stochastic modelling.

There are established methods for DA price forecasting, and commercialsoftware can do the task. Short-term RT price forecasts are used by manymarket participants, too. However, we do not know any such solutions forlong-term RT price forecasting. In this work we build a RT price modellingmethod that uses simulated DA price time series and several other inputvariables. A study that introduces a forecasting method for AS price is done


in [16].

2.3.1 Day-ahead price forecasting

DA price creation is well-known on fundamental level and there are severalmethods to create forecasts. Commercial software products exist for short-term and long-term forecasting [16], [17], [3]. electricity market participantscan use them supporting their decision making [32].

For more information on DA price forecasting models see e.g. [34], [32],[4], or [23].

In this work we do not focus on DA price forecasting, but we use simulatedDA price series as input for our RT price forecasting method. It is possibleto forecast DA price accurately enough to be used as an input for stochasticRT price forecasting model.

2.3.2 Real-time price forecasting

Fundamental models for short-term forecasting of RT price exist and are usedby market participants. However, methodologies to forecast RT price reliablymany years to future do not exist (Rhodri Williams, Regional director - ER-COT, Genscape, Boston, 10 February 2016). Generally RT price forecastingis more difficult than forecasting DA price, as RT price is more volatile (shortbut very high spikes) and impacted by many more variables. In this work webuild a stochastic method to create simulated RT price forecasts for severalfuture years.

2.4 ERCOT

2.4.1 Overwiev

ERCOT market covers about 90 % of Texas load and serves 24 million con-sumers [12]. Installed generation capacity of ERCOT is more than 74 GWand current record demand, 69.6 GW, occurred on August 10 2015. ERCOTis a nodal energy-only market, which means that generating companies re-ceive all their revenues from selling energy and AS products, not from theiron-line capacity. There is a separate RT price, locational marginal price(LMP), for each node. LMPs differ from each other when congestion occurs[11]. So called hub prices are average prices of node LMPs. We will forecasthub average price in this study, which is the simple average of all ERCOThub prices.


As mentioned earlier, ERCOT is almost an electric island due to lim-ited capacity of interconnections with adjacent markets. This transmissioncapacity is only 1.3 GW [9].

Figure 2.2 shows shares of installed generation capacity in ERCOT in2014 by fuel type. 55 % of generation capacity is natural gas -fired. Coalpower plants account for 24 % and nuclear plants for 6 %. Wind accounts for14 % of installed capacity and wind generation capacity continues to growin future years [9]. Solar capacity is less than 1 % of installed capacity.

Figure 2.2: Shares of generation capacities by fuel in 2014 in ERCOT [12].

Therefore, we do not treat it separately in this study. However, the trend isincreasing in ERCOT [9], which means that impact of solar generation mayneed to be added to price forecasting models in future. Since solar generationhas very low marginal production cost, but is intermittent by nature, it canbe supposed that addition of solar generation will lower prices on sunnydays. On the other hand, solar generation can be supposed to increase pricevolatility by increasing magnitude of net-load changes and need to dispatchfast-ramping power plants.


2.4.2 Different markets of ERCOT

In ERCOT, there are two energy-only markets: DA energy market and RTenergy market. For ancillary services (AS) there is only DA market. Anoverview to market procedures of each market is provided in [11]. A thoroughdescription of all market rules and processes can be found in ERCOT marketrules and protocols, e.g. [14] and [10].

2.4.2.1 ERCOT Day-ahead energy market

DA market clearing is performed every morning at 10 a.m. for electricitydelivery of the next day. DA market is hourly. Participating in DA marketis voluntary for generators. The generators provide for each hour their three-part supply offers, specifying their start-up offer (USD they want for startingto generate), minimum-energy offer (minimum stable load and respectiveprice), and energy offer curve (price-quantity pairs above minimum stableload). Market is cleared by ERCOT as least-cost solution. This means thatERCOT optimises the system based on all offers from generators so that thegeneration equal to load will be carried out with least cost possible to theelectricity consumers [11].

2.4.2.2 ERCOT Real-time energy market

RT market clearing is made every five minutes. Gate closure is five minutes,which means that actual electricity delivery happens five minutes after mar-ket clearing. Generators provide their three-part supply offers, as they do inDA market. Market clearing is done as least-cost solution [11]. Volume ofRT market in MW is lower than that of DA market. However, prices in DAmarket and longer-term forward markets that are bigger in volume closelyfollow RT price, which makes financial impact of RT market bigger than itsMW-volume suggests [29].

2.4.2.3 ERCOT Day-ahead ancillary services market

In ERCOT, there are four AS products: regulation up, regulation down,responsive reserve, and non-spinning reserve. Like DA energy market, ASmarket is cleared at 10 a.m. for the next day. AS market is hourly [15].

Chapter 3

Statistical analysis of RT price

3.1 Analysis of RT electricity price and its

potential drivers

We will next study historical market data published by ERCOT and collectedand delivered to us by Genscape. The goal is (i) to build quantitative under-standing of drivers of RT price and (ii) to choose an appropriate simulationmethodology to create future time series of RT price.

3.1.1 DA price

As explained in chapter 2, DA price DA is determined by market clearingexecuted by ERCOT on basis of bids and offers sent by market participants.Bidding behaviour of market participants is largely impacted by expectationsof probabilities of specific conditions occurring on the next day (in personalcommunication with Kevin Hanson, Supervisor, Market Operations Support,ERCOT, 8 February 2016, Taylor TX).

Figure 3.1 shows DA price from the beginning of year 2011 to the endof year 2015. Price is most of the time less than 100 USD/MWh. Medianprice is 27.44 USD/MWh. The highest prices are in summer of 2011, around2500 USD/MWh. Periods when price is above 500 USD/MWh are rare.Such periods have occurred less than ten times in years 2011-2015. Eventhough there are only a few high spikes in DA price, they have a signifi-cant financial impact on electricity sellers and buyers. Duration curve is agraph commonly used to illustrate empirical distribution of electricity price.Vertical axis represents prices from the smallest to highest in the period ofinterest. Horizontal axis represents the hours of year (8760 in total). Itcan be scaled if the inspection period length differs from one year. Then it

28

CHAPTER 3. STATISTICAL ANALYSIS OF RT PRICE 29

Figure 3.1: Day-ahead price from year 2011 to 2015. Most of the timeDA price is below 100 USD/MWh, but occasional spikes of more than 100USD/MWh occur.

represents average hours of year in the inspection period. Duration curveshows how many hours in a year (value on horizontal axis) electricity pricewas above any price level (value on vertical axis). Duration curve is alwaysdecreasing. Exactly same information could be shown in empirical cumula-tive distribution function or histogram (though infinite number of intervalswould be needed in histogram in general case). Duration curve of DA pricein figure 3.2 shows that highest DA price in 2011-2015 has been around 2600USD/MWh and lowest has been approximately zero. DA price has exceeded100 USD/MWh only a few hundred hours per year. Figure 3.3 shows DAprice and demand of an example week in June 2014. Both variables showclear seasonal variation at one day season length. Values of both variables areoften low at night and high in the day-time. Demand of last two days of theinspection period is higher than in the other days. DA price has reached itsmaximum in these days, too. DA price is largely impacted by demand sinceprice is determined by the dispatched power plant with the most expensiveoffer price. In low-demand times it is enough to dispatch renewable and baseload plants with cheap offer prices. As demand increases intermediate andpeaking plants with higher offer prices need to be dispatched to serve thedemand. Consequently price increases.


Figure 3.2: Duration curve of DA price in top figure shows that only a tinyshare of DA prices was above 500 USD/MWh. Bottom figure shows partof duration curve where DA < 150. Price is below 50 USD/MWh approxi-mately 8000 hours per year. Almost linearly decreasing duration curve showsthat price distribution is rather uniform between 0 and 50 USD/MWh.

3.1.2 RT price

ERCOT has specified a price cap, which limits how high the RT price RTcan be. Table 3.1 shows historical levels of price cap as specified in [6].Figure 3.4 shows RT price from the beginning of year 2011 to the end ofyear 2015. Price is most of the time less than 100 USD/MWh. Median RTprice is 25.2 USD/MWh. Cap prices have occurred unregularly in the five-year period and comparison with figure 3.1 shows that high RT prices aremore frequent than high DA prices. Because of greater height and frequency,the financial impact of price spikes on market participants is even larger inRT than DA market. However, as the DA price is hourly and RT price is5-minute price, the duration of RT spikes can be, and often is, a lot shorterthan in DA. A histogram of historical durations of price spikes is shown infigure 3.5. Majority of spikes lasts only 5 minutes and only a tiny share ofspikes is longer than 3 hours (36 5-minute periods). A power plant that cancapture the value of short price spikes by being able to ramp up and downvery fast, can make significant profit in RT market in addition to its DAmarket profit [30]. Majority of the highest RT prices are in the year 2011,on cap level 3000 USD/MWh. The reason for high prices in the summer


Figure 3.3: Top graph shows DA price of an example week in June 2014 andbottom graph shows demand in the same period. Both time series have aclear seasonal variation at one day season length. Price is often high, whendemand is high, and opposite.

of 2011 was very hot weather in Texas leading to very high demand and togeneration outages [27]. The price spikes in February 2011 are a result ofgeneration outages and problems with gas equipment caused by very coldweather [27]. In RT price not only prices above 1000 USD/MWh, but alsoprices between 500 and 1000 USD/MWh are more frequent than in case ofDA price. These prices are mainly due to quick net-load changes that requireGTs to be started, since there is not enough cheaper fast-ramping powerplant generation capacity available (in personal communication with KevinHanson, Supervisor, Market Operations Support, ERCOT, 8 February 2016,Taylor TX). GTs have high cost per start and the need to dispatch themincreases the price to a high level for at least the first 5-minute period thatthey are running.

Duration curve of RT in figure 3.6 shows that highest price in 2011-2015 has been approximately 5000 USD/MWh and lowest has been negative,approximately -250 USD/MWh, where also price floor is located. RT hasexceeded 100 USD/MWh only a few hours per year. Excluding both endsof duration curve, the rest is very stable. This means that more than 8500hours a year, price has been inside a small interval. As spikes are so rare, wecan model them separately in the forecasting method.


Figure 3.4: Real-time price from year 2011 to 2015. Most of the time priceis low, but there are sometimes spikes. Year 2011 shows most huge pricespikes of 3000 USD/MWh. Prices between 500 and 1000 USD/MWh aremost frequent in the period starting at second half of 2012.

RT is strongly positively autocorrelated on short lags. This can be seenin figure 3.7, showing the empirical autocorrelation function of RT . At lag100, i.e. 8 hours, autocorrelation has almost vanished already, but there issome positive 24-hour seasonal autocorrelation, around lag 288. This is dueto daily variation that was seen even more clearly in DA and is also presentin RT . To account for the strong autocorrelation of RT price, we can use theprice of previous time step to explain the next price.

Figure 3.8 shows RT and DA of an example week in June 2014. DAmoves seasonally up and down by time of day and RT follows it to some

Table 3.1: ERCOT price cap 2011-2015Start date Price capN/A 30001 Aug 2012 45001 Jun 2013 50001 Jun 2014 70001 Jun 2015 9000


Figure 3.5: Histogram of durations of historical price spikes. More than 400of total 600 spikes lasts only 5 minutes (one time step). The longest pricespike in 2011-2015 has been almost 6 hours (70 5-minute periods) long.

extent. However, RT movements are not as regular as those of DA. highestRT price is often lower than highest DA price of the day. This systematicdifference between RT and DA can be seen as a risk premium. It representsthe expected profit loss that risk-averse electricity buyers are willing to acceptin DA market to avoid having to buy their electricity in RT market whereprice volatility is higher and consequently risk of price spike is higher. Loadis low at night and unexpected events occur less often than in the day-time. Consequently risk of price spike is lower at night. Therefore, risk-premium is lower at night (in personal communication with Kevin Hanson,Supervisor, Market Operations Support, ERCOT, 8 February 2016, TaylorTX). An estimate of risk premium can be obtained from mean differenceof RT and DA, which is 2.97 USD/MWh. Risk premium estimated in thesame way using only prices that have occurred between 3 p.m. and 4 p.m.is a lot higher, 19.65 USD/MWh. This difference results from the higherspike probability of RT price spike in the afternoon. In addition, there aretwo short spike periods in RT in the example week. The risk of extremelyhigh price realized for electricity buyers that have not bought their electricity


Figure 3.6: Duration curves of DA and RT price in top graph shows that hugemajority of both prices are low. Huge prices of above 1000 USD/MWh andnegative prices are rare. Bottom graph shows part of duration curve whereprice is in range 0-100 USD/MWh. It can be seen that DA prices in range40-100 USD/MWh are slightly more frequent than RT prices in the samerange. On the contrary, RT prices below 40 USD/MWh are more frequent,and especially very low prices are relatively more frequent in RT than in DAmarket.

from DA market. Figure 3.9 shows empirical joint distribution of DA andRT . There is no clear linear or non-linear correlation visible. However, vastmajority of points in the graph is close to origin, which makes it virtuallyimpossible to draw conclusions from the graph. It is possible that there is adependency between DA and RT , when both are at their most common level.A zooming to prices below 100 USD/MWh is shown in figure 3.10. Thereis clear linear correlation between DA and RT when DA is between 0 andaround 40 USD/MWh and then the two prices are often almost equal. WhenDA is higher than 40 USD/MWh, RT price distribution is more uniform and


Figure 3.7: Empirical autocorrelation function of RT price shows strong pos-itive autocorrelation on lags shorter than 50 time-steps (5-minute periods).There is seasonal autocorrelation at lags near 288, i.e. 24 hours.

RT is more often below DA. This results from the risk premium relatingto DA and RT markets described above. DA is often almost equal to RT ,when there is little uncertainty related to RT . In such situations prices areoften low. When uncertainty of RT is greater, DA is more often above 40USD/MWh and RT does not correlate so strongly with DA.


Figure 3.8: DA and RT prices of last week of June 2014. RT price does notshow as clear daily pattern as DA price. There are two RT price spikes inthe period. Highest DA price of day is often above the highest RT price ofday.

Figure 3.9: Scatter plot of DA price and RT price. Vast majority of pointsare close to origin. RT prices above 1000 USD/MWh have occurred at alllevels of DA price, but their relative share is greater when DA price exceeds500 USD/MWh. Negative RT prices have only occurred when DA price hasbeen at low level.


Figure 3.10: Scatter plot of DA price and RT price when both are below 100USD/MWh. Grey dashed line represents situation when RT and DA pricesare equal. RT price clearly increases as DA price increases between 10 and40 USD/MWh. Systematic increase of RT price with DA price stops whenDA price exceeds 40 USD/MWh.


3.1.3 Spread between RT and DA price

We denote the spread between DA and RT price by ∆ = RT −DA. Figure3.11 shows ∆ of years 2011-2015 as a function of time. There are largepositive ∆ values in the same periods that RT has spiked. Due to price floorRT can be significantly below DA only when DA is high. Therefore, largenegative ∆ values can occur only at periods that DA is high. As high DAprices have been rare, there are only a few large negative ∆ values in theperiod.

Figure 3.11: ∆, i.e. spread between RT and DA price from year 2011 to2015. Most of the time ∆ is close to zero. Occasional large negative andpositive deltas occur. Majority of them are positive.

Duration curve of ∆ in figure 3.12 shows that the highest ∆ value in2011-2015 was around 5000 USD/MWh and lowest ∆ value was around -3000 USD/MWh. Absolute value of ∆ exceeded 100 USD/MWh only a fewhours per year. Duration curve is flat when both ends are excluded. Thismeans that more than 8500 hours a year, spread between DA and RT hasbeen very small. Mean ∆ is −2.97 USD/MWh, meaning that on average RTis that much lower than DA.

Empirical autocorrelation function of ∆ is shown in figure 3.13. ∆ exhibitsstrong positive autocorrelation on lags shorter than 50 time-steps (5-minuteperiods). There is a positive seasonal autocorrelation at lags near 288, i.e.24 hours, much stronger than in case of RT . This is possibly a result of risk


Figure 3.12: Duration curve of ∆ shows that vast majority of ∆ values areclose to zero, but tails of ∆ distribution are very long.

premium of DA regularly widening in high-load day-time and getting smallerin the night-time.

Scatter plot of DA price and ∆ in figure 3.14 shows clear negative corre-lation between the two variables. This is obvious, resulting from the way wedefined ∆ as RT − DA. Vast majority of points are close to origin. Largepositive ∆ values are possible only when DA price is low enough due to pricecap. Similarly, large negative ∆ values are possible only when DA price ishigh enough due to price floor. As we saw in figure 3.10, RT follows DAclosely when DA is in its most common level of less than 40 USD/MWh.Then ∆ is often close to zero and small unexpected events in market canmake it slightly negative or positive. The uniform distribution of ∆ meansthat DA being below 40 USD/MWh, RT price simulation can be carried outby first simulating DA and then sampling ∆ and adding it to RT . However,simulation of RT can not be conducted in the same way when DA is above 40USD/MWh since distribution of ∆ is strongly dependent on DA as we sawin figure 3.14. In that region, however, distribution of RT is rather uniformas seen in figure 3.9. Therefore, we can simulate RT price by sampling RT


Figure 3.13: Empirical autocorrelation function of ∆ shows strong positiveautocorrelation on lags shorter than 50 time-steps (5-minute periods). Thereis clear positive seasonal autocorrelation at lags near 288, i.e. 24 hours.

directly, when DA ≥ 40.

3.1.4 Demand

Figure 3.15 shows demand of electricity in ERCOT from the beginning ofyear 2011 to the end of year 2015. The demand varies between 20000 and70000 MW. There is clear yearly seasonal variation in demand because ofthe greater need for air-conditioning in summer and for warming houses inwinter. There might also be an increasing trend as the demand seems toincrease slightly towards the end of the period. Furthermore, there is a clearseasonal variation in demand on daily basis as we saw in figure 3.3. Loadincreases in the morning. Night-time load is often significantly lower thanin the day-time. Figure 3.16 shows a duration curve of demand. It can beseen that demand being over 60000 MW is rare, but maximum demand isaround 70000 MW. This means that upper tail of demand distribution islong. To be able to serve the demand in the highest demand times even ifsome outages occur, system needs more than 10000 MW of capacity, thatis unused most of the year. Seasonal qualities at different lengths described


Figure 3.14: Scatter plot of DA price and ∆ shows clear negative correlationbetween the two variables. Vast majority of points are close to origin. Largepositive ∆ values have occurred only when DA price is low enough due toprice cap. Similarly, large negative ∆ values have occurred only when DAprice is high due to price floor.

above can be seen in the empirical autocorrelation function of demand, too,shown in figure 3.17, where clear peak is seen around lags 288 (=24 hours)and its multiples. This results from the fact that demand is usually closeto what it was 24 hours earlier. Autocorrelation on short lags is close toone, meaning that demand changes between consecutive 5-minute periodsare often rather small.

Top graph of figure 3.18 shows scatter plot of demand and RT price. Itseems that demand does not explain RT price movements. High prices arerather evenly distributed at all demand levels excluding lowest demand valuesof the period. Slightly more cap prices occur when demand is close to itsmaximum than otherwise. Bottom graph shows scatter plot of demand and∆. Like high RT values, high ∆ values have occurred evenly at all demandlevels. However, large absolute values of ∆ are a lot more frequent whendemand is high, over 60000 MW. These are caused by demand being almostequal to total available generation capacity. Price increases naturally whenmore expensive generation has to be dispatched, but also through price-


Figure 3.15: ERCOT demand 2011-2015 varies between 20000 and 70000MW. There is clear yearly seasonal variation, summer-time demand beinghigher than rest of the year. There are also some high-demand periods inwinter-time, especially in years 2011, 2014, and 2015.

adder, that was deployed in ERCOT in 2014. Price-adder increases pricewhen surplus capacity, i.e. available generation capacity exceeding demandis less than 5 GW [13]. A zooming to most common ∆ level in figure 3.19shows that distribution of ∆ is rather uniform at its normal levels conditionalon demand. Therefore, we do not need demand as an explanatory variableof ∆. Most high RT prices that have occurred at high-demand times are nota result of high demand itself, but rather of low surplus capacity.

Scatter plot of day-ahead forecast and actual demand in figure 3.20 showsvery strong positive correlation between the two variables. This means thatdemand forecast errors are small most of the time. Sometimes, though, actualvalue differs slightly from forecast. Both ends of point cloud are thinner thanmiddle part. It suggests that forecasts are often more accurate when demandis close to its annual maximum or minimum. However, this may as well be aresult of natural variation and a lot more points being in the middle of cloud.


Figure 3.16: Duration curve of demand shows that there are less than 500hours per year in which demand exceeds 60000 MW. Majority of hours,demand is between 30000 and 50000 MW.

Figure 3.17: Autocorrelation function of demand shows very strong autocor-relation at lags shorter than 20 time-steps, i.e. around two hours. There isvery strong seasonal 24-hour autocorrelation.


Figure 3.18: Top graph shows scatter plot of demand and RT price. No clearcorrelation is visible. However, there are slightly more high RT prices whendemand is close to its maximum due to cap prices of the year 2011. Bottomgraph shows scatter plot of demand and ∆. Variance of ∆ is higher whendemand is high.


Figure 3.19: Scatter plot of demand and ∆ limited to −30− 30 USD/MWh,its most common level. There is no clear correlation between the two vari-ables.


Figure 3.20: Scatter plot of forecast and actual demand shows very strongpositive correlation between the two variables. Both ends of the cloud areslim, which tells us that very strong and very weak wind conditions are easiestto forecast.


3.1.5 Wind generation

Rapid increasing trend in installed wind generation capacity makes studyingimpact of wind generation on RT price very important for the purpose ofour study. If we could not capture the impact accurately enough, therewould be a good chance that our forecast series would not exhibit true futurecharacteristics of future ERCOT RT price.

Figure 3.21 illustrates ERCOT wind generation from year 2011 to 2015.Top graph shows that wind generation varies between close to zero and morethan 10000 MW. There is an increasing trend as a result of growing installedwind generation capacity. Bottom graph shows wind generation as a per-centage of installed capacity. For example, 100% would mean that all windgenerating units are generating at their full capacity. There is no clear trendand each year the values have varied from close to zero to a little over 80%. A mean of all these values, capacity factor, is 32 %, which tells that onaverage each wind generating unit has run on 32 % of its maximum outputin 2011-2015. This number sounds reasonable, as it is a lot lower than 100%, which would require idealistic perfect wind conditions and no mainte-nance breaks at generators. Top graph of figure 3.22 shows duration curve2011-2015 of wind generation in MW. It shows that generation of more than8000 MW has occurred only 1000 hours per year on average. However, thisgraph is not very informative due to rapid growth of installed wind gener-ation capacity. Bottom graph shows duration curve of wind generation asa percentage of installed capacity. We saw in figure 3.21 that there is notrend in percentage values. Therefore, percentage values give more constis-tent information about wind conditions. Output exceeding 80 % of installedcapacity is rare (100 hours per year) and zero generation situations occurhardly ever. Distribution between these extremes is rather even. Empiricalautocorrelation function of wind generation on top graph of figure 3.23 showsvery strong autocorrelation at lags shorter than 50 time-steps, i.e. aroundfour hours, because of slowly moving weather fronts. There is also a weakseasonal 24-hour component in autocorrelation possibly due to regular dailyvariation of wind speed in Texas. Bottom graph shows empirical autocorre-lation function of error of day-ahead forecast of wind generation. It exhibitsstrong short-term autocorrelation, too, but not as strong as wind generation.There is no clear 24-hour peak in the graph. As we may expect, accuracyof forecasts made for same time of two consecutive days do not correlate.Top graph of figure 3.24 shows scatter plot of wind generation and RT price.There seems to be no clear correlation between the two variables. High RTprices, though, occur a little more frequently, when wind generation is lowthan otherwise. Bottom graph shows scatter plot of wind generation and


Figure 3.21: Top graph shows wind generation from 2011 to 2015. It varies alot between 0 and more than 10000 MW. There is an increasing trend visible.The highest values are in the end of year 2015 due to windy conditionsand largest installed wind generation capacity. Bottom graph shows windgeneration as a percentage of installed capacity. There is no clear trend andeach year the values have varied from close to zero to a little over 80 %.

∆. There is no clear correlation visible. Absolute value of ∆ is almost al-ways very small, when wind generation exceeds 10000 MW. Scatter plot ofhistorical day-ahead forecast made by ERCOT and actual wind generationin figure 3.25 shows that the values of the two variables often differ only alittle. It means that wind generation forecasts are relatively accurate mostof the time. A few points are far from the cloud, which means that actualwind generation sometimes differs significantly from forecast. The ends ofthe cloud are slim. Forecasts are generally more accurate, when there is verystrong or hardly any wind, than when wind speed is normal. This is pos-sibly a result of the usual shape of generation curve of a wind generator asfunction of wind speed. Generation increases rapidly at intermediate windspeeds causing more variance in generation output, when it is at its middlelevel [22].


Figure 3.22: Top graph shows duration curve of wind generation in MW. Itshows that generation of more than 8000 MW has occurred only 1000 hoursper year on average. The highest values are even more rare. Bottom graphshows wind generation as a percentage of installed wind generation capacity.Wind generation exceeds 30 % of installed capacity on average 4000 hoursper year. Zero generation situations occur hardly ever.


Figure 3.23: Empirical autocorrelation function of wind generation on topgraph shows very strong autocorrelation at lags shorter than 50 time-steps,i.e. around four hours. There is also seasonal 24-hour component in auto-correlation. Bottom graph shows empirical autocorrelation function of windgeneration forecast error. It exhibits strong short-term autocorrelation, too,but not as strong as wind generation. There is no clear 24-hour peak in theautocorrelation function.


Figure 3.24: Top graph shows scatter plot of wind generation and RT price.No clear correlation is visible. However, there are slightly more high RTprices when wind generation is close to zero. Bottom graph shows scatterplot of wind generation and ∆. There are no clear correlations either. ∆ isalmost always very small in absolute value, when wind generation exceeds10000 MW.


Figure 3.25: Scatter plot of day-ahead forecast and actual wind generationshows positive correlation between the two variables, indicating that windgeneration forecasts are relatively accurate most of the time. There aresome points far from the middle of the cloud, which means that actual windgeneration sometimes differs significantly from forecast. The ends of thecloud are slim, which shows that forecasts are generally more accurate, whenthere is very strong or hardly any wind, than when wind speed is normal.


3.1.6 Net-load

Figure 3.26 shows a scatter plot of net-load and RT on top graph. There isno clear correlation between the two variables. High prices occur at all levelsof net-load except the lowest net-loads of less than 20000 MW. Bottom graphshows scatter plot of net-load and ∆. It seems that net-load does not explainoccurrence of high ∆ values very well. Largest negative ∆ values, however,have occurred at very high net-load times. Possibly DA is high becauseof high probability of high RT due to forecast surplus capacity being low.However, sometimes the risk does not realize and RT is consequently low.

When net-load increases rapidly, only fast-ramping power plants can re-spond to the change. Some of these power plants, especially GTs, havehigh marginal generation cost. Need to dispatch them leads to high electric-ity prices. Therefore, we can expect to see positive correlation in net-loadchange speed and RT . Top graph of figure 3.27 shows scatter plot of 5-minute net-load change, NLC, and RT . NLC is positive when net-load isincreasing. Clearly, there are more high RT values at positive than negativenet-load change values. Bottom graph shows scatter plot of NLC and ∆.Large negative ∆ values are more frequent when NLC is negative. Similarly,large positive ∆ values are more frequent when NLC is positive. The clearestlimit for ∆ is not NLC = 0, but NLC = 200 MW. We can use NLC beingabove or below 200 MW to explain ∆ distribution.


Figure 3.26: Top graph shows scatter plot of net-load and RT price. No clearcorrelation is visible, but it seems that RT price has always been close to zerowhen net-load has been below 20000 MW. Bottom graph shows scatter plotof demand and ∆. Large negative ∆ values are more frequent, when net-loadis high.


Figure 3.27: Top graph shows scatter plot of 5-minute net-load change andRT price. Positive value means growing net-load. High prices are morefrequent when net-load change is positive. Bottom graph shows scatter plotof net-load change and ∆. Large negative ∆ values are more frequent whennet-load change is negative and large positive ∆ values are more frequentwhen net-load change is positive.


3.1.7 Non-intermittent generation

Actual available non-intermittent generation capacity, NICA, represents themaximum net-load that could be served with available generation resources.Generators submit their current operating plans (COP) to ERCOT. COPssubmitted for next day can be used as day-ahead forecast of available capac-ity, NICF . Figure 3.28 shows scatter plot of NICF and NICA. The graphshows very strong positive correlation between the two variables, indicatingthat last-day changes to COPs are small most of the time. There are severalpoints far from the rest of points, which means that sometimes moderate de-viations from forecast have occurred. These deviations are possibly a resultof unexpected power plant outages.

Figure 3.28: Scatter plot of day-ahead forecast and actual values of avail-able on-line non-intermittent generation capacity shows very strong positivecorrelation, indicating that forecast errors are very small most of the time.There are several points far from the rest of the cloud, which means thatsometimes moderate changes from forecast occur.


3.1.8 Surplus capacity

ERCOT implemented an Operating Reserve Demand Curve (ORDC) on June1, 2014 to incentivize building new generation capacity. ERCOT InvestmentIncentives and Resource Adequacy in 2012 had showed that system reliabilitymight be threatened otherwise [27], [13]. As a result, scarcity pricing mecha-nism increases RT price when available reserves are below 5000 MW. Scatterplot of actual surplus capacity FCA = NICA + IGA − LA and RT pricein figure 3.29 shows that spike prices mostly occur when surplus capacity islow. Surplus capacity represents the amount by which available generationcapacity exceeds load. This is partly a result of scarcity pricing mechanismand partly of power plants with higher offer prices being dispatched.

We can use amount of surplus capacity as an explanatory variable forprice spike probability in our simulation methodology.

Figure 3.29: Scatter plot of surplus capacity and RT price shows that highprices mostly occur when surplus capacity is low. Share of high RT prices isa lot higher, when surplus capacity is below 7000 MW.


3.1.9 Forecast error of surplus capacity

We define forecast error of surplus capacity, FCFE, as difference between ac-tual surplus capacity FCA and day-ahead forecast FCF , FCFE = FCA−FCF , where FCA = NICA+ IGA−LA and FCF = NICF + IGF −LF .Therefore, FCFE represents the difference between actual surplus capacityand day-ahead forecast of surplus capacity, which was available when DAmarket was cleared. Negative FCFE means that actual surplus capacity islower than was expected day-ahead. Figure 3.30 shows histogram of FCFE.Most of the time FCFE is between −5000 and 2000 MW.

Figure 3.30: Histogram of forecast error of surplus capacity. Positive valuemeans that actual surplus capacity is larger than day-ahead forecast. His-togram shows that forecast slightly underestimates actual surplus capacitymost of the time.

We expect that RT is often higher than DA, when FCFE is negative,since there is less extra resources available than was expected. Figure 3.31shows one example night from June 2014, when a price spike occurred be-cause of largely negative FCFE. Top graph shows actual and forecast windgeneration. It can be seen that after 22:00 actual generation drops rapidlyeven though it was forecast to increase slightly. Middle graph shows forecasterror of surplus capacity in the same period. It falls sharply from close tozero to −8000 MW. At the same time there is a huge spike in ∆, shown inthe bottom graph. This occurred six days after ORDC was implemented andprice-adder was one factor that increased RT price.

Figure 3.32 shows scatter plot of forecast error of surplus capacity and RTprice on top graph. There is no clear correlation between the two variables.Bottom graph shows scatter plot of forecast error of surplus capacity and∆. Large negative values of ∆ occur only, when forecast error of surplus


Figure 3.31: A graph of wind generation of one example night in June 2014shows how the generation unexpectedly drops rapidly from 8000 MW to lessthan 3000 MW. Middle graph shows that forecast error of surplus capacitydrops consequently to -8000 MW. Bottom graph shows that a large negative∆ has occurred at the same time.

capacity is positive and large positive values of ∆ are more frequent, whenforecast error of surplus capacity is negative. Therefore, we can use forecasterror of surplus capacity to explain ∆ movements. Clearest difference in∆ distribution is seen when FCFE is divided into two classes with limitFCFE = −1800 MW.


Figure 3.32: Top graph shows scatter plot of forecast error of surplus capacityand RT price. Positive value on horizontal axis means that actual surpluscapacity is larger than day-ahead forecast. There is no clear correlationbetween the two variables. Bottom graph shows scatter plot of forecast errorof surplus capacity and ∆. Large negative ∆ values are more frequent, whenforecast error of surplus capacity is positive. Large positive ∆ values are alittle more frequent, when forecast error of surplus capacity is negative.


3.1.10 Node prices and hub prices

We have used as RT price in analyses the hub average RT price, which issimple average of hub prices in ERCOT. Next we will briefly study differencesin prices of different nodes and hubs. Each power plant can choose whetherit sells its electricity for node price or hub price (in personal communicationwith M.Sc. Matti Rautkivi, Origination, Wartsila North America, 8 Febru-ary 2016, Taylor TX). Therefore, it is important to know what impact onprofitability the selection of reference price has.

Figure 3.33 shows that congestion causes larger differences in RT pricebetween hubs (bottom graph) than within a hub (top graph). This is justone example analysis and a lot more node prices should be analysed to buildcomprehensive understanding of general differences in hub and node prices.We will do simulations in this study using hub average price. If future pricesof a specific node are needed, it is safest choice to do simulations for the priceof that node. It is simple as only one input time series needs to be changed.However, there is a risk that adding new generation or transmission capacityto that node has an impact on future prices. Therefore, hub price is moreconservative choice, as it is more robust with respect to local changes in thegrid.

3.2 Selection of simulation methodology

As expected, our analysis did not show perfect correlation between RT priceand any other variable. However, we found some patterns that explain apart of RT price variation. Because of no perfect correlations and significantfuture uncertainty that is present in our simulations we can use a stochasticmethod in RT price simulation. Due to large variation in RT price we di-vide it to several price classes. We build a modified bootstrap model whereprice class is sampled first and then price from the right class. Explanatoryvariables have impact on sampling probabilities of price classes. Since DAmarket modelling is relatively straightforward and accurate, we can use DAmarket modelling software to create simulated DA price and DA-forecastsof explanatory variable values as inputs for our RT price modelling method.We will sample ∆, when DA < 40 and RT otherwise. Stochasticity and im-pact of explanatory variables are applied to selection of price class for eachtime-step.

We simulate spike prices separately and define spike period loosely asa period in which RT price is mostly in it highest percentile. A precisedefinition will be given later in the study. We will use the surplus capacity


Figure 3.33: Comparison of example hub prices and node prices. Top graphshows RT price of Houston hub on horizontal axis and RT price of one nodein Houston hub on vertical axis. Prices differ only a little from each other.Prices of hub average and West hub, shown on bottom graph, differ moreoften than intra-hub prices on top graph.

FCA and DA price DA to determine spike starting probability with limits7000 MW and 30 USD/MWh, respectively.

We also found that RT price being on non-spike level, variation can beexplained by forecast error of surplus capacity FCFE, change speed of net-load NLC, DA price DA, and price class of previous time-step. Limits for


Table 3.2: Division points of variables used in non-spike situations.FCFE -1800 MWNLC -200 MWDA [25,40] USD/MWhRT (t− 1) [16.7, 19.1, 21.3, 22.9, 24.5, 26.2, 28.8, 32.6, 39.8]∆(t− 1) [−13.7,−8.6,−6.0,−4.2,−2.7,−1.5,−0.4, 0.7, 2.4]

explanatory variables FCFE,NLC, and DA that make clear difference inprice distribution are shown in table 3.2. Table also shows limits of priceclasses and delta classes determined as deciles of hub average RT price and∆.

Strong autocorrelation exhibited by RT price is supposedly obtained insimulated time series as a result of strongly autocorrelated explanatory vari-ables, by taking into account the price of previous time-step when samplingprices, and by sampling prices for spike periods as blocks of historical spikeprices instead of sampling each price separately.

Chapter 4

Stochastic RT price simulation

In this chapter we introduce a simulation method and use it to obtain futuretime series of RT price. The main idea of our simulation methodology isto determine historical correlations between the explanatory variables anddependent variable RT price, and then use these correlations together withsimulated future values of explanatory variables in stochastic simulation offuture RT price. By correlation we do not mean Pearson correlation co-efficient or any other specific statistic, but any quantity that can be usedto statistically measure the association between two variables. The mostcommonly used measure of correlation in our simulation is (empirically esti-mated) conditional probability.

First we divide the historical RT price data into price classes and estimateempirical probabilities of price belonging to each price class. Then we sim-ulate future time series of explanatory variables using DA market modellingsoftware and stochastic methods. Finally, we conduct a stochastic simula-tion of RT price based on simulated explanatory variables and estimatedprobabilities.

As explained in Chapter 3, we can divide the time steps to two subsets byDA. When DA < 40, we will simulate ∆, i.e. the spread between DA andRT . Otherwise we simulate directly RT . To take into account the impactof explanatory variables on RT price, we introduce a concept of dynamictime step category C, which is calculated for all time steps. In our modellingapproach C is the reason behind different price patterns occurring in differentconditions. Not all variation of RT price can be explained by C, but we haveused all information that was determined useful in our analysis. Biggestweakness of our simulation approach that is based on bootstrap method isimperfect prediction accuracy. It results from the wide intervals in whichthe explanatory variables can vary within a dynamic time step category.Assuming that the values of explanatory variables are precise, accuracy could

64

CHAPTER 4. STOCHASTIC RT PRICE SIMULATION 65

be improved by using for example some regression method. However, ourapproach enables us to create thousands of future RT price time series, andindividual errors caused by random selection of values can be assumed tocancel in large number of simulations. Secondly, the future time series ofexplanatory variables are far from precise. Therefore, greater accuracy inprediction would not help to improve accuracy of RT price prediction. For athorough discussion of bootstrap methods in time series simulation, see [5].

4.1 Calculating needed historical inputs

We start our simulation by dividing the historical RT prices RT into groupscorresponding to nRTC price classes GRTC = {RT |RT ∈ RTC}, RTC =0, 1, ..., nRTC −1 and historical deltas ∆ = RT −DA into groups correspond-ing to n∆C

delta classes G∆C= {∆|∆ ∈ ∆C},∆C = 0, 1, ..., n∆C

−1. A simpleway to determine delta classes and price classes is to use p∆% and pRT% per-centiles of historical data, respectively, where p∆ = 100

n∆C

and pRT = 100nRTC

. For

example, if n∆C= 4, we will have all ∆ values below the first quartile of ∆

in ∆C = 0, values of ∆ below median, but above the first quartile in ∆C = 1,etc. The prices of time steps t with DA(t) ≥ DAlimits,C(1) are divided intoprice groups GRTC , and deltas of time steps t with DA(t) < DAlimits,C(1) aredivided into delta groups G∆C

.We group the values of explanatory variables into variable dynamic time

step category C = 0, 1, ..., n(C), which is calculated separately for each timestep and includes all needed information about the conditions that applyduring time step t and previous price class or delta class. The largest C,Cspike corresponds to spike time step. Definitions of all dynamic time stepcategories C for n∆C

= nRTC = 12 are shown in table 6.1 in appendix.When we have determined C(t) and ∆C(t) or RTC(t) for each historical

time step (5-minute period) t, we can calculate empirical conditional prob-ability distributions of (i) ∆ belonging to each ∆C , P (∆C |C), and similarlyof (ii) RT belonging to each price class RTC , P (RTC |C) conditional on Cof that time-step. Conditional probabilities are calculated in the usual wayby dividing the number of time-steps with each C and ∆C or RTC by totalnumber of periods with that C. The probability distribution of delta classesbecomes

P (∆C |C) =n(∆C ∩ C)

n(C), C 6= Cspike,

where n(x) is the number of periods in the history data with the quality x.


The probability distribution of price classes becomes

P (∆C |C) =n(∆C ∩ C)

n(C), C 6= Cspike.

Because of huge variation in RT and DA, we treat periods of spiky priceseparately. We build a vector GRTspike containing the prices of spike periodsin a chronological order. We also determine empirical probabilities of spikestarting conditional on the amount of surplus capacity FCA, and DA priceDA. These probabilities will be used later in simulation of future spikeperiods.

To give a precise definition of spike period, we first specify a limit lspike,RT ,above which all RT are spikes, and a similar limit lspike,DA for DA. Wedenote by binary Xspike each time step being spike (Xspike = 1) or non-spike (Xspike = 0). If RT or DA (or both) of time-step t exceeds the limit,RT (t) ≥ lspike,RT or DA(t) ≥ lspike,DA, a spike period starts. The spike periodlength is determined as the longest possible L such that the share of spikeprices in the spike period is at least Sspike, a percentage parameter specifiedin advance.

Exact description of methodology is shown in algorithm 1.

Algorithm 1 Historical correlations

Data Historical time series of RT price and explanatory variablesRTt = Real-time priceDAt = Day-ahead priceIGAt = Intermittent generation, actualNICAt = Available non-intermittent generation capacity, actualLAt = Load, actualIGFt = Intermittent generation, forecastNICFt = Available non-intermittent generation capacity, forecastLFt = Load, forecastNLCt = Net-load change from previous time step


OutputGrouped RT prices, deltas, and spike lengthsG∆C

= {∆hist|∆hist ∈ ∆C} = Historical ∆ values of delta class ∆C ∈{0, 1, 2, ..., n∆C

− 1}GRTC = {RThist|RThist ∈ RTC} = Historical RT prices of price class

RTC ∈ {0, 1, 2, ..., nRTC − 1}GRTspike = [RThist(t)|Xspike,hist(t) = 1] = Historical RT prices of spike

periods in chronological orderGL = Lengths of historical spikesEstimated probabilitiesP (∆C |C) Estimated probabilities of delta classes ∆C ∈

{0, 1, 2, ..., n∆C− 1} conditional on dynamic time step category C

P (RTC |C) Estimated probabilities of price classes RTC ∈{0, 1, 2, ..., nRTC − 1} conditional on dynamic time step category C

P (Xstart|IDAlimit,prob(DA), IFCAlimit,prob

(FCA)) Estimated probabilitiesof spike starting conditional on day-ahead price and actual surplus capacity

Parameterslspike,RT limit above which all RT prices are spike priceslspike,DA limit above which all DA prices are spike pricesDAlimit,prob = Division point of DA for conditional probability of spike

period startingFCAlimit,prob = Division point of FCA for conditional probability of

spike period startingSspike = Share of time steps that must have spike price within a spike

periodlRTC = vector that contains limits of price classesl∆C

= vector that contains limits of delta classesDAlimits,C = vector that contains limits of DA for determining CLhistory = number of historical time stepsFCFElimit = limit of FCFE for determining CNLClimit = limit of NLC for determining C

procedureSpike periodsXspike ← 1− (1− Ilspike,RT

(RT ))(1− Ilspike,DA(DA))

t← 1while t ≤ Lhistory do . Go through all history data

C0(t) = 4[IDAlimits,C(1)(DA(t)) + IDAlimits,C(2)(DA(t))] +2IFCFElimit

(FCFE(t)) + INLClimit(NLC(t))


if Xspike(t) = 0 then . Non-spike time stepif t > 1 then

if Xspike(t− 1) = 1 thenC(t) = C0(t) + 12(n∆C

+ nRTC )else

C(t) = C0(t) + 12[∆C(t − 1) + IDAlimits,C(1)(DA(t −

1))[n∆C+RTC(t− 1)−∆C(t− 1)]

]end if

else . First time step

C(t) = C0(t) + 12[∆C(t) + IDAlimits,C(1)(DA(t))[n∆C

+

RTC(t)−∆C(t)]]

end ifif DA(t) ≥ DAlimits,C(1) then

RTC(t) = IlRTC(1)(RT (t)) + IlRTC

(2)(RT (t)) + ... +IlRTC

(nRTC−1)(RT (t))

GRTC(t) ← [GRTC(t), RT (t)]else

∆C(t) = Il∆C(1)(∆(t))+Il∆C

(2)(∆(t))+ ...+Il∆C(n∆C

−1)(∆(t))G∆C(t) ← [G∆C(t),∆(t)]

end ift← t+ 1

else . Spike period startsfor all τ = 1, 2, ..., Lhistory − t+ 1 do

S(τ) =Xspike(t)+Xspike(t+1)+...+Xspike(t+τ−1)

τ

end forLs ← min(τ |S(τ) < Sspike)− 1 . Last consecutive time step

with share of spike prices large enoughtd ← 0while td ≤ Ls do . Cut off non-spike prices from end

if Xspike(t+ Ls − 1− td) = 0 thentd ← td + 1

elseL← Ls − tdXspike(r), r = t, t+ 1, ..., t+ L− 1← 1td ←∞

end ifend whileGL ← [GL, L] . Save length and pricesGRTspike ← [GRTspike , RT (t), RT (t+ 1), ..., RT (t+ L− 1)]t← t+ L . Go to the end of this spike and continue

end ifend while

end procedure


4.2 Creating future time series of explana-

tory variables

We will use actual values of the explanatory variables and day-ahead forecastsof them to create forecast electricity price scenarios. In particular, we willuse the difference between the simulated day-ahead forecast and simulatedactual value of several explanatory variables (non-intermittent generationcapacity, wind generation, demand) to determine the probability of pricebeing on different levels. One possible way to create future time series ofexplanatory variables would be simulating future actual values first, and thenbootstrapping forecast errors from historical forecast errors to achieve futureforecast values of explanatory variables. However, since we can simulatethe forecast values using commercial electricity market modelling software,we will do the same procedure in the opposite order. We simulate forecastvalues first and then we simulate actual values of explanatory variables.

It is important to understand that we do not try to predict what thedemand and values of other explanatory variables will be each hour of thenext years in ERCOT. Instead, we will create several different forecast timeseries, none of which is likely to be precisely accurate. However, all simulatedtime series will approximately exhibit the statistical quantities and seasonalpatterns that have occurred in history.

4.2.1 Day-ahead market simulation

As explained in Chapter 3, there are relatively accurate modelling methodsfor forecasting DA price, when several explanatory variables are known. Wecan use a commercial modelling software to future DA price. The same soft-ware can be used to create day-ahead forecasts of demand and generation,based on historical forecast data and estimate of future changes in severalmeaningful market quantities such as development of ERCOT wind gener-ation capacity. As DA market modelling is relatively straightforward andestablished methods exist, we focus in this study on creating real-time sce-narios based on simulated DA prices and forecasts. We take DA electricityprice and day-ahead forecasts of demand and generation as given. How-ever, it is possible that simulated time series would be biased as a result ofrandomness in DA market simulation. We will carry out several stochasticsimulations of explanatory variables to reduce impact of such bias on resultsof our RT price simulation.

The outputs of DA market simulation that we will use as inputs for ourRT market simulation are processes of day-ahead forecasts of intermittent


generation IGFt, demand LFt, non-intermittent generation capacity NICFt,and DA electricity price DAt.

4.2.2 Simulating actual values of explanatory variablesthrough Markov chain and bootstrap methods

Figure 4.1: Classes of day-ahead forecast are separated by black verticallines. Different colours represent the groups of actual values separated byconditional quantiles. Within each interval separated by two vertical lines,the ten colours correspond to deciles of values of intermittent generation inthat interval.

Forecast values obtained using DA market modelling software are usedas a starting point when we create future time series of actual values of theexplanatory variables. We use bootstrap method to choose an actual valuecorresponding to each forecast value. To avoid incredibly large differencesbetween simulated day-ahead and actual values, we divide the forecast values


into equally wide classes XFC . We sample each actual value from the histor-ical actual values occurred when the historical forecast was in the same class.To preserve the strong autocorrelation of actual values in history, we dividethe historical actual values occurred while XF was in class XFC to q setsGXAXFC,QXA

, XFC ∈ {0, 1, 2, ..., nXFC− 1}, QXA ∈ {1, 2, ..., q} of equal num-

ber of observations. We simulate the future groups QXA so that historicalautocorrelation is preserved. We use Markov chain approach with transitionmatrix estimated from historical observations. Then we simply sample theactual value XA(t) from the set corresponding to group QXA and class XFC .

Figure 4.1 shows the groups of IGA separated by conditional quantilesQXA = 1, ..., q = 10 (colours). QXA = 1 corresponds to 10% conditionalquantile, QXA = 2 corresponds to 20% conditional quantile, etc. Blackvertical lines represent limits of classes XFC .

Next, we define the algorithm that can be used to simulate future actualvalues of any variable X. We will use this algorithm for IGA, NICA, andLA. Simulation methodology is shown in algorithm 2.


Algorithm 2 Actual future values of explanatory variablesData

Processes of variable X, i.e. the variable of interestXAhist. = historical actual values of XXFhist. = historical day-ahead forecast values of XXFfut. = future day-ahead forecast values of X

OutputXAfut. = Simulated future scenario of X

Parametersq = number of groups separated by conditional quantiles per XFC

used in simulationnXFC

= number of equally wide intervals XF is divided intolXFC

= vector of limits of XFClXAXFC

= vector of conditional quantiles of XA conditional on XFCLhistory = number of historical time stepsLfuture = number of time steps to simulate

procedureGrouping values of XAhist.for all t = 1, 2, ..., Lhistory do

XFC(t)←∑nXFC

−1

j=1 IlXFC(j)(XFhist.(t)) . Class of forecast

QXA(t)←∑q−1

k=1 IlXAXFC (t)(k)(XAhist.(t)) . Conditional quantile

GXAXFC (t),QXA(t)← [GXAXFC (t),QXA(t)

, XAhist.(t)] . Actual valuesaved to right group

end forEstimating empirical transition probabilities between groupsP (QXA(t)|QXA(t− 1))← n(QXA(t)∩QXA(t−1))

n(QXA(t−1))

Simulating XAfut.for t = 1 do

Sample group QXA(t) with equal probabilities for all q groupsSample XA(t) from QXA(t) with equal probabilities

end forfor all t = 2, 3, ..., Lfuture do

Sample group QXA(t) with probabilities P (QXA(t)|QXA(t− 1))

XFC(t)←∑nXFC

−1

j=1 IlXFC(j)(XFfut.(t))

XAfut.(t) sampled from GXAXFC (t),QXA(t)with equal probabilities

end forend procedure


4.2.2.1 Wind generation

New wind generation units are often built in same windy regions as theexisting ones. Therefore, their output fluctuations are highly similar as windspeed changes. The difference between forecast and actual output of onewind generating unit is strongly correlated with another that is situatedin the same region. Therefore, output powers of units can be treated asstrongly positively correlated random variables. Thus, it can be assumed thatincreasing wind generation capacity will result in forecast errors increasinglinearly in the same proportion.

To account for this linear impact of new-build generation capacity ontotal output and forecast errors we divide historical IGAhist. and IGFhist.values by historical system-wide installed wind generation capacity of eachtime step. We obtain forecast and actual wind generation of each time stepas percentages IGFhist. − % and IGAhist. − %, respectively, of installed ca-pacity. We also simulate future day-ahead forecast by DA market modellingsoftware as percentage of installed capacity IGFfut. − %. Now the simula-tion algorithm for actual values used with inputs XFhist. = IGFhist. − %,XAhist. = IGAhist. − % and XFfut. = IGFfut. − % gives also actual windgeneration time series as percentage of installed capacity IGAfut. − %. Totransform it into MW-value we can simply multiply by an estimate of futureinstalled capacity. This provides also a natural way to study the impact ofincreasing or decreasing installed wind generation capacity on RT price. Wecan create scenarios with same IGAfut. −%, but different installed capacityand compare simulated RT price time series.

4.2.2.2 Available non-intermittent generation capacity

In case of available non-intermittent generation capacity forecast errors aremainly due to changes in plans of generating companies. They may be aresult of e.g. unexpected outages or changes in near-term demand fore-casts that are used as inputs for decision making of power plant operation(Rhodri Williams, Regional director - ERCOT, Genscape, 10 February 2016,Boston MA). We assume that the forecast errors of non-intermittent genera-tion capacity will be linearly dependent on the total installed capacity of non-intermittent generating units. Therefore we can use a similar methodology tocreate future time series of actual available non-intermittent generation ca-pacity NICAfut. as we used for actual wind generation IGAfut.. Obviously,values are not calculated as percentages of installed wind generation capacity,but of installed non-intermittent generation capacity. Data for the simula-tion algorithm is then XFhist. = NICFhist. − %, XAhist. = NICAhist. − %


and XFfut. = NICFfut.−%. The output time series NICAfut.−% is actualavailable non-intermittent generation capacity time series as percentage of in-stalled capacity. It can be transformed into absolute MW-value in the sameway as in case of wind generation by multiplying by an estimate of futureinstalled capacity. We can also study the impact of increasing or decreasinginstalled non-intermittent generation capacity on RT price by creating sce-narios with same NICAfut.−%, but different installed capacity and comparesimulated RT price time series.

4.2.2.3 Demand

Since forecast errors of demand mainly result from errors in weather forecasts,we can assume that forecast errors will be linearly dependent on annual peakload, i.e. the highest demand of a year. We can use the same method to createfuture series of actual demand, LAfut., that we use for actual wind genera-tion IGAfut. and available non-intermittent generation capacity NICAfut..Input values are calculated as percentages of annual peak load. Data for thesimulation algorithm is then XFhist. = LFhist.−%, XAhist. = LAhist.−% andXFfut. = LFfut. −%. The output series LAfut. −% is actual demand timeseries as percentage of annual peak load and it can be transformed into ab-solute MW-value by multiplying by an estimate of annual future peak load.Again, we can study the impact of increasing or decreasing demand on RTprice by creating scenarios with same LAfut. − %, but different peak loadgrowth rate, and compare simulated RT price time series.

4.3 Stochastic simulation methodology for fu-

ture RT price

We use a stochastic simulation methodology to create simulated future RTprice time series. First we simulate prices for future spike periods, and thenwe simulate prices of rest of the time steps. We use groups GRTC and G∆C

corresponding to RT and ∆ values of each price class and delta class fromsimulation 1 as the populations from which future RT prices and ∆ valuesare sampled. Price class or delta class of each future time step is selectedrandomly using estimated conditional probabilities of classes conditional onvalues of explanatory variables and the price class or delta class of previoustime step. Therefore, the simulation model could be classified as a dynamicbootstrap method.


4.3.1 Simulating RT prices of future spike periods

We start by sampling a binary variable Xstart for each future time step indi-cating whether a spike period starts on that time step. Sampling is performedwith conditional probabilities determined in 1. Then we sample duration ofeach spike period, Lspike, from a group of historical spike lengths GL. Theprices for spike periods are obtained using block sampling method. Instead ofsampling each price separately, we sample as many blocks of length LB withreplacement until all prices of the spike period are sampled. This methodhelps to get the autocorrelation of sampled time series as strong as in history.A small correction to keep the distribution of prices of spike periods sameas historical is done in block sampling by using wrapping method. In thismethod a block can start near the end of and ”continue” in the beginning ofvector GRTspike .

Algorithm 3 Spike simulationData

Simulated processes of explanatory variablesDAt = Day-ahead priceIGAt = Intermittent generation, actualNICAt = Non-intermittent generation capacity, actualLAt = Demand, actualSampling populationsGRTspike = RT prices of historical spike periods in chronological orderGL = Lengths of historical spike periodsEstimated probabilitiesP (Xstart|IDAlimit,prob

(DA), IFCAlimit,prob(FCA)) Estimated probabilities

of spike starting conditional on DA and FCA


ParametersLB = Price block lengthDAlimit,prob = Division point of DA for conditional probability of spike

startingFCAlimit,prob = Division point of FCA for conditional probability of

spike startingLfuture = number of future time steps to simulate

procedure Simulationt← 0while t ≤ Lfuture do

FCA(t)← NICA(t) + IGA(t)− LA(t)Xstart(t) chosen randomly with P (Xstart|IDAlimit,prob

(DA), IFCAlimit,prob(FCA))

if Xstart = 1 thenLspike sampled from GL

RTtemp ← Lspike values chosen randomly from GRTspike usingblock sampling method with block length LB with wrapping

for all τ = t, t+ 1, ..., t+ Lspike − 1 doXspike(τ)← 1RT (τ) = RTtemp(τ − t+ 1)

end fort← t+ Lspike

elset← t+ 1

end ifend while

end procedure


4.3.2 Simulating other RT prices than spike periods

Now that we have simulated prices of future spike periods, we can simulateprices of non-spike time steps. Dynamic time step category C is determinedfirst based on values of explanatory variables. Price class, if DA ≥ DAlimits,C ,or delta class, otherwise, is sampled using class probabilities conditional onC. Then RT or ∆ is sampled from chosen class with equal probabilities.Simulation method for non-spike time steps is shown in algorithm 4.

Algorithm 4 Non-spike simulationData

Simulated processes of explanatory variablesDAt = Day-ahead priceIGAt = Intermittent generation, actualNICAt = Non-intermittent generation capacity, actualLAt = Load, actualIGFt = Intermittent generation, forecastNICFt = Non-intermittent generation capacity, forecastLFt = Load, forecastXspiket = Binary vector telling whether each time step is spike or notSampling populationsG∆C

= {∆hist|∆hist ∈ ∆C} = Historical ∆ values of delta class ∆C ∈{0, 1, 2, ..., n∆C

− 1}GRTC = {RThist|RThist ∈ RTC} = Historical RT prices of price class

RTC ∈ {0, 1, 2, ..., nRTC − 1}GRTspike = [RThist|Xspike,hist = 1] = RT prices of historical spike peri-

ods in chronological orderEstimated probabilitiesP (∆C |C) Probabilities of delta classes ∆C ∈ {0, 1, 2, ..., n∆C

− 1} con-ditional on dynamic time step category C

P (RTC |C) Probabilities of price classes RTC ∈ {0, 1, 2, ..., nRTC − 1}conditional on dynamic time step category CParameters

DAlimits,C = vector that contains limits of DA for determining CFCFElimit = limit of FCFE for determining CNLClimit = limit of NLC for determining Cn∆C

= number of delta classesnRTC = number of price classes


procedure Simulationfor t = 0 do

∆C(0) chosen randomly with equal probabilities∆(0) chosen randomly from G∆C(0) with equal probabilities

end forfor all t > 0 do

C0(t) = 4[IDAlimits,C(1)(DA(t)) + IDAlimits,C(2)(DA(t))] +2IFCFElimit

(FCFE(t)) + INLClimit(NLC(t))

if Xspike(t) = 0 thenif Xspike(t− 1) = 1 then . Previous was spike

C(t) = C0(t) + 12(n∆C+ nRTC )

if DA(t) ≥ DAlimits,C(1) thenRTC(t) chosen randomly with probabilities P (RTC |C(t))RT (t) chosen randomly with equal probabilities from

GRTC(t)

else∆C(t) chosen randomly with probabilities P (∆C |C(t))∆(t) chosen randomly with equal probabilities from

G∆C(t)

RT (t) = DA(t) + ∆(t)end if

else . Previous was not spike

C(t) = C0(t)+12[∆C(t−1)+IDAlimits,C(1)(DA(t−1))[n∆C

+

RTC(t− 1)−∆C(t− 1)]]

if DA(t) ≥ DAlimits,C(1) thenRTC(t) chosen randomly with probabilities P (RTC |C(t))RT (t) chosen randomly with equal probabilities from

GRTC(t)

else∆C(t) chosen randomly with probabilities P (∆C |C(t))∆(t) chosen randomly with equal probabilities from

G∆C(t)

RT (t) = DA(t) + ∆(t)end if

end ifend if

end forend procedure


4.4 Validating model functioning by compar-

ing historical and simulated values

4.4.1 Actual values of explanatory variables

We can create simulated time series of actual values of explanatory vari-ables LA, IGA, and NICA using historical forecast values of LF , IGF , andNICF as inputs. If time series simulated in this way are similar to theirhistorical correspondents, we can conclude that our simulation methodologycan be used to create simulated future scenarios based on forecast valuessimulated by commercial DA market modelling software. We conduct threeseparate simulations, one for each variable, with same parameters in eachcase. We divide the historical data to nXFC

= 50 intervals each 2% widecompared to maximum value of forecast. We divide actual values to q = 10groups, i.e. conditional deciles of each 2 % interval of forecast. Exampleweeks of all three variables are shown in figures 4.2, 4.3, and 4.4. Thereare differences between historical and simulated actual values. However, thedifferences are so small that they can be supposed to be a result of randomvariation that is natural and desired in stochastic simulation.

Figure 4.2: Black solid line shows actual historical load of Christmas week2015. Blue dashed line shows actual load simulated using historical inputs.Simulated and historical demand are almost exactly equal.

Historical and simulated duration curves for each variable are shown infigure 4.5. They coincide almost perfectly, meaning that distribution of sim-ulated values is highly similar to that of historical.

Another measure for performance of our wind generation simulation iscapacity factor, i.e. mean of wind generation as percentage of nameplate


Figure 4.3: Example week of actual wind generation. Black solid line rep-resents historical actual wind generation of Christmas week 2015. Blue linerepresents actual values simulated with historical inputs. The movements ofthe two lines are highly similar. Perfect matching is not desired in stochasticsimulation.

Figure 4.4: Example week of actual non-intermittent generation capacity.Black line represents actual historical values of Christmas week 2015 andblue line represents simulated actual values with inputs of year 2015.

capacity. Historical capacity factor of years 2011-2015 is 32 %. Histogramof capacity factors of 300 simulation runs in figure 4.6 shows that they aregenerally almost exactly equal to historical.

Analysis of explanatory variables in Chapter 3 showed that they all ex-hibit strong autocorrelation. Top row of figure 4.7 shows empirical autocorre-lation functions of historical and bottom row of simulated actual values. All


Figure 4.5: Duration curves of explanatory variables. Black solid lines repre-sent actual historical values of 2015 and blue dashed lines represent simulatedactual values with inputs of year 2015. Simulated duration curves are veryclose to their historical correspondents.

three simulated variables exhibit similar simple and seasonal autocorrelationas historical.

The three graphs on left side of figure 4.8 show scatter plot of historicalforecast and actual values of load, intermittent generation, and availableon-line non-intermittent generation capacity. Graphs on the right side showsimilar graphs of simulated values. Simulated time series of all three variablesexhibit similar strong positive correlation between forecast and actual valueas historical series do.

Comparisons between historical and simulated actual values of demand,wind generation, and non-intermittent generation capacity show very closeresemblance. Based on analysis, we can conclude that our stochastic sim-ulation methodology for actual values of explanatory variables is sufficient.Next we need to confirm that other parts of simulation method are valid,too. After that we can create simulated RT price scenarios.


Figure 4.6: Histogram of capacity factors of 300 simulation runs of actualwind generation. Black vertical line represents historical capacity factor 32%. Simulated capacity factors are close to historical. Small differences aredesired and result from randomness in simulation.


Figure 4.7: Empirical autocorrelation functions of historical (top) and simu-lated (bottom) LA, IGA, andNICA. Simulate series imitate autocorrelationstructures of their historical correspondents almost perfectly.


Figure 4.8: Scatter plots of DA-forecast and actual values of historical (left)and simulated (right) demand, LF and LA, wind generation, IGF and IGA,and available non-intermittent generation capacity, NICF and NICA. Con-ditional distributions of actual values as functions of forecasts closely resem-ble those of historical values.


Table 4.1: Parameters used in algorithm 1lspike,RT 93.19 USD/MWhlspike,DA 300 USD/MWhDAlimit,prob 30 USD/MWhFCAlimit,prob 7 GWSspike 0.5lRTC [16.7, 19.1, 21.3, 22.9, 24.5, 26.2, 28.8, 32.6, 39.8]l∆C

[−13.7,−8.6,−6.0,−4.2,−2.7,−1.5,−0.4, 0.7, 2.4]DAlimits,C [25, 40]FCFElimit -1800 MWNLClimit 200 MWLhistory 525600

4.4.2 RT price

To ensure that the model described above can be used to create future RTprice time series, we apply it with historical input data and compare theoutput series with actual historical occurred series. If simulated RT pricetime series resemble historical price, we can conclude that the model functionsas expected. All market data that we use was collected by ERCOT andprocessed and delivered to us by Genscape.

Historical data that we use for simulation includes historical time se-ries of DA and RT , explanatory variables LA, LF , IGA, IGF , NICA,NICF , and time series of installed wind generation capacity and installednon-intermittent generation capacity from the years 2011-2015. Individualmissing data points and periods of missing data with length less than orequal to two hours are filled by linearly interpolating. In case of time seriesof dependent variables DA and RT interpolation is not used. As changes inthe values of explanatory variables from time step to time step are often rel-atively small, linear interpolation is a safe choice for short periods of missingdata. For large periods of missing data, interpolation is not used. Graphs ofinput variables as a function of time are shown in chapter 3. All parametersused in this simulation are shown in table 4.1.

Algorithm 1 run with inputs and parameters as specified, gives us theempirical conditional probabilities P (∆C |C) of delta classes and price classesas functions of dynamic time step category C. Explanations of all dynamictime step categories C in case of 12 price classes and delta classes are shownin table 6.1 in appendix. An example of conditional probabilities of deltaclasses are shown in table 4.3 and conditional probabilities of price classesare shown in table 4.4. Conditional spike starting probabilities are shown in


Table 4.2: Conditional probabilities of price spike at different values of ex-planatory variables DA price DA and actual surplus capacity FCA. Allprobabilities are rather low, but at FCFE values above 7000 MW (rightcolumn) probabilities are a lot lower.

FCFE (MW)DA (USD/MWh) < 7000 ≥ 7000< 30 0.002 0.0001≥ 30 0.006 0

table 4.2.Now we have determined all needed quantities from historical data and

can start to simulate future RT price. In this validation we do not carryout DA market simulation or simulation of actual values of explanatoryvariables to obtain simulated series of DA, LA, LF , IGA, IGF , NICA,and NICF since we do not want randomness in these simulations to im-pact our simulated RT price. Therefore, we use historical values of ex-planatory variables as inputs for RT price simulation. In addition we usegrouped history data G∆C

, GRTC , GRTspike , GL and conditional probabili-ties P (Xstart|IDAlimit,prob

(DA), IFCAlimit,prob(FCA)), P (∆C |C), and P (RTC |C)

output in the previous step by algorithm 1 for algorithms 3 and 4. The num-ber of time steps to simulate, Lfuture, is 1051200, i.e. 10 years. Parametervalues used for spike simulation are shown in table 4.5 and values used forsimulation of non-spike RT prices are shown in table 4.6.

One simulated week of RT price and the corresponding historical RTprice are shown in figure 4.9. Behaviour of RT prices is highly similar. Itis clear that the spike periods do not coincide and it is not desirable in ourstochastic simulation approach. There are a few price spikes in both historicaland simulated RT price. Spikes occur mainly when DA price is close to itsdaily maximum. There are also short periods when RT price is close to zero.In this example week such situations only occur at nights when demand islow and stable. Our simulated RT price seems to preserve this quality well.Visual inspection does not show large differences in autocorrelation betweenthe price series.

Median of historical RT price is 25.23 and median prices of 100 simulationruns are shown in histogram in figure 4.10. Medians are very near to eachother, again indicating that simulated price series resemble the historicalprice series. It is not desirable in our stochastic approach that median pricewould be precisely same.

Duration curves of historical and simulated RT price are shown in figure4.11. They coincide almost perfectly. Small differences are natural and de-


Table 4.3: Example of estimated conditional probabilities of delta classesP (∆C = i|C) conditional on dynamic time step categories C 0-7 and 12-19.Majority of dynamic time step categories is excluded from this example.

delta class iC 0 1 2 3 4 5 6 7 8 90 0.43 0.09 0.01 0.03 0.06 0.1 0.07 0.09 0.07 0.041 0.28 0.04 0.03 0.04 0.15 0.1 0.09 0.1 0.04 0.12 0.64 0.09 0.02 0.03 0.05 0.04 0.05 0.02 0.03 0.033 0.23 0.06 0.03 0.09 0.09 0.14 0.11 0.03 0.23 04 0.48 0.1 0.07 0.06 0.07 0.06 0.03 0.03 0.03 0.065 0.54 0.14 0.08 0.04 0.02 0.04 0.04 0.01 0.01 0.086 0.55 0.07 0.09 0.07 0.05 0.03 0.03 0.03 0.03 0.067 0.52 0.12 0.04 0.09 0 0.06 0.01 0.04 0.04 0.0912 0.07 0.64 0.1 0.05 0.02 0.04 0.02 0.02 0.02 0.0213 0.1 0.8 0.1 0 0 0 0 0 0 014 0.11 0.68 0.09 0.03 0.03 0.02 0.02 0.01 0.01 015 0.09 0.45 0.14 0.05 0.05 0.14 0.09 0 0 016 0.02 0.86 0.09 0.02 0.01 0 0 0 0 017 0.01 0.87 0.09 0.01 0 0 0.01 0 0 018 0.02 0.85 0.08 0.03 0.01 0.01 0 0 0 019 0.03 0.81 0.16 0 0 0 0 0 0 0


Table 4.4: Example of estimated conditional probabilities of price classesP (RTC = i|C) conditional on dynamic time step category C ∈{8, 9, 10, 11, 20, 21, 22, 23, 32, 33, 34, 35, 44, 45, 46, 47, 56, 57, 58, 59}. Majorityof dynamic time step categories is excluded from this example.

price class iC 0 1 2 3 4 5 6 7 8 98 0 0 0 0 0.03 0.02 0.03 0.22 0.33 0.379 0 0.05 0 0 0.15 0.05 0.1 0.05 0.3 0.310 0 0.01 0.01 0.01 0.02 0.02 0.04 0.08 0.29 0.5411 0 0 0 0.08 0.08 0 0 0.15 0.23 0.4620 0 0.02 0 0 0.15 0.26 0.39 0.17 0 021 0 0 0 0.14 0 0 0.43 0.43 0 022 0 0 0.03 0.05 0.16 0.08 0.51 0.16 0 023 0 0 0 0 0.5 0 0 0.5 0 032 0 0 0 0.04 0.04 0.07 0.21 0.61 0.04 033 0 0 0 0 0.06 0.06 0.22 0.5 0.17 034 0 0 0 0 0.04 0.2 0.28 0.44 0.04 035 0 0 0 0 0 0 0.14 0.43 0.43 0

Table 4.5: Parameters used in spike simulation.DAlimit,prob 30 USD/MWhFCAlimit,prob 7 GW

Table 4.6: Parameters used in simulation of non-spike RT prices.DAlimits,C [25, 40] USD/MWhFCFElimit -1800 MWNLClimit 200 MWn∆C

10nRTC 10


Figure 4.9: Example week. Movements of historical (black solid line) andsimulated (blue dashed line) RT price are highly similar. They do not co-incide perfectly and, in fact, it is not desirable in our stochastic simulationapproach.

sired result of randomness in simulation. Distribution of simulated RT pricematches very closely with that of historical RT price.

Empirical autocorrelation functions of both RT price time series are shownin figure 4.12. It can be seen from the two graphs on the left that the strongseasonal autocorrelation of historical RT price on lags approximately 288 (24hours) is not exhibited by simulated series. However, the graphs on the rightshow that autocorrelation of simulated series on short lags resembles that ofhistorical price, being only slightly weaker on lags 1-10. For the purpose ofprofitability calculation for a flexible power plant it does not have any impactthat the long lag autocorrelation is not preserved by our simulation. Fast-ramping power plants can change their operating plans many times within aday and they are not dependent on events of previous or next day.

From all graphs and quantities compared above we can conclude that oursimulation methodology can create simulated time series of RT price highlysimilar to historical series in all qualities and quantities that are important


Figure 4.10: Histogram of median prices from 100 simulated scenarios usinghistorical values of input variables. Median prices are generally close tohistorical median price 25.23 USD/MWh.

for the purpose of our simulation. Next, we will use the simulation methodto study the impact of increasing relative share of wind generation capacityas a percentage of demand on RT price in ERCOT.

4.5 Case studies

Next we will use our forecasting model to create simulated future RT pricetime series. We consider two cases: (i) base scenario, where all importantquantities develop as predicted by ERCOT or stay stable, and (ii) growthscenario where installed wind generation capacity and annual peak load growslightly faster. We run 100 stochastic simulations of each scenario usingexactly same historical inputs, RT and DA prices and explanatory variables.We only vary the future values of installed wind generation capacity and


Figure 4.11: Duration curves of historical (black solid line) and simulated(blue dashed line) RT price match almost perfectly.

annual peak load. Installed non-intermittent generation capacity is assumedto develop similarly in both scenarios. We compare our simulation resultsof both scenarios to reveal any differences in RT price behaviour caused bydifferences in input variables.

4.5.1 Base scenario and growth scenario

Our historical explanatory variable values consist of ERCOT market data2012-2015 published by ERCOT, processed and delivered to us by Genscape.Data includes historical time series of DA and RT , explanatory variables LA,LF , IGA, IGF , NICA, NICF , and time series of installed wind generationcapacity and installed non-intermittent capacity. Individual missing datapoints and periods of missing data with length less than or equal to two hoursare filled by linearly interpolating, except in case of time series of dependentvariables DA and RT . Interpolation method is same that we used earlier


Figure 4.12: Both historical (top) and simulated (bottom) RT price exhibitstrong autocorrelation on short lags (right). Simulated price, however, doesnot exhibit as strong seasonal autocorrelation (left) on lags approximately288 time steps (24 hours).

in validation of model functioning. As representations of RT and DA priceswe use hub average price, which is, as explained earlier in chapter 3, simpleaverage of all hub prices in ERCOT. Parameters used are same as those usedfor validation runs shown in tables 4.1, 4.5, and 4.6.

Possible differences between scenarios can only result from differences insimulation of future values of explanatory variables LA, LF , IGA, IGF . Weconsider two scenarios that slightly differ from each other. As explained ear-lier, our simulation methodology for future values of explanatory variablesenables us to create different scenarios of demand and wind generation devel-opment. We simply change the time series of peak load and installed windgeneration capacity that we use to multiply the simulated values of windgeneration as a percentage of installed capacity and demand as a percentageof annual peak load.

In base scenario we use stable installed wind generation capacity as inputof future development. Installed wind generation capacity grows from 18 GWin 2016 to 23 GW in 2017 and stays stable thereafter. In growth scenario,capacity grows 8 % per year faster. Growth speed is still slow compared to 12% geometric mean annual growth in 2011-2015. Difference between capacity


in the two scenarios grows gradually from zero in 2016 to 28 GW in 2025.Growth scenario is more realistic in installed wind generation capacity growth(8 % vs. 0 %, historical 12 % ) but it is still slightly conservative estimate.However, the difference between scenarios is important in our simulation, notabsolute values. Graphs of development of installed wind generation capacityin both scenarios are shown in figure 4.13.

Figure 4.13: Installed wind generation capacity grows faster in growth sce-nario (dashed blue line) than in base scenario (solid black line). Differencegrows from zero in 2016 to 28 GW in 2025.

ERCOT estimates that peak load is 70 GW in 2016 and grows thereaftersteadily to 78 GW in 2025. We use this estimate in our base scenario. Thegeometric mean of yearly peak load growth will be 1.2 % per year in 2016-2025 in base scenario. We modify that number to 3 % in growth scenario,which means that peak load in 2025 is 91 GW. Graphs of development ofpeak load in both scenarios are shown in figure 4.14. It can be seen thatgrowth of peak load is stronger in growth scenario than in base scenario.Difference grows gradually from zero in the year 2016 to 13 GW in 2025.

Growth scenario has both demand and installed wind generation capac-


Figure 4.14: Peak load growth in growth scenario (dashed blue line) is fasterthan in base scenario (solid black line). Peak-load difference between scenar-ios grows from zero in 2016 to 13 GW in 2025.

ity growing faster than in base scenario. Growth scenario will have greaterpeak load, but also greater installed wind generation capacity. Higher de-mand, other things being equal, naturally increases electricity price as thereis less surplus capacity and power plants with higher offer prices need tobe dispatched. On the other hand, higher wind generation can be expectedto decrease prices. These two differences between base and growth scenariocan be expected to cancel impact of each other on RT price to some extent.However, there may be some differences in simulation results, e.g. as a resultof more frequent and larger net-load ramps caused by intermittent nature ofwind generation.

4.5.2 Comparison of results in different scenarios

100 simulations were run of both scenarios and three example RT price timeseries of both scenarios for years 2016-2025 are shown in figure 4.15. Theyresemble each other visually, but high prices are slightly more frequent in


growth scenario RT price time series.

Figure 4.15: Three simulated RT price series of base scenario (left) are mostlysimilar to series of growth scenario (right). Growth scenario time series havea few more 9000 USD/MWh cap prices.

Histogram of median prices of the 100 simulation runs for each scenarioare shown in figure 4.16. Median prices of growth scenario are consistently alittle higher than median prices of base scenario simulation runs. However,difference is only a few cents per MWh. Duration curves of RT prices of all200 simulation runs are shown in figure 4.17. There are no large differencesin graphs. Figure 4.18 shows only 200 most expensive hours of year and onlyprices below 1000 USD/MWh. The dashed blue lines representing durationcurves of growth scenario are clearly higher than solid black lines of base sce-nario. For example, RT price in base scenario simulations is only 40-70 hours


above 400 USD/MWh, whereas RT price in growth scenario simulations is100-160 hours above 400 USD/MWh.

Figure 4.16: Median RT prices of 100 simulation runs of both scenarios showslight difference between scenarios. Median prices of growth scenario aregenerally a little higher (21.53-21.63 USD/MWh) than median prices of basesimulation runs (21.47-21.56 USD/MWh). Difference in median prices is sosmall that it has practically no impact on market participants.

The differences in duration curves suggest that the impact of wind gener-ation capacity increasing faster in growth scenario creates more price spikesin RT market. Increased number of high-price hours without increase in av-erage price means that price volatility is greater in growth scenario than inbase scenario.

Systematic differences in the simulated RT price time series of differentscenarios result from differences in demand and wind generation capacity. Wecan conclude that increasing wind generation capacity share of all generationcapacity increases volatility of RT price in ERCOT. This increase in volatilitymay give opportunities for flexible power plants to sell energy in RT marketand benefit from transactions between DA and RT markets.


Figure 4.17: 200 duration curves, 100 of each scenario, show hardly anyvariation. The graphs are almost similar, except in the highest-price hours(left end).


Figure 4.18: Duration curves of 100 simulation runs of base scenario andgrowth scenario. Blue dashed lines are clearly higher than black solid lines,indicating that high prices are more frequent in growth scenario.

Chapter 5

Conclusions

The purpose of this work was to understand real-time (RT) price creationin ERCOT market and build a simulation methodology that can be usedto create simulated RT price series for several future years. We conductedstatistical analyses to identify price drivers and constructed a simulationmethodology using bootstrap method with several modifications.

We began with a short introduction to electricity markets and took acloser look at ERCOT market. Next, we examined the impact of differentfactors on RT price and selected explanatory variables to be used as inputsof our forecasting model based on results of statistical analysis. Future val-ues of the chosen explanatory variables can be simulated by conducting aday-ahead (DA) market simulation using a dedicated commercial software.We introduced a stochastic model to create simulated RT price time series.We validated its functioning by using historical inputs and comparing out-put RT price series to historical RT price. Moreover, we carried out futuresimulations to compare RT price in two future scenarios, (i) base scenario inwhich installed wind generation capacity is stable and peak-load develops infuture as predicted by ERCOT and (ii) growth scenario in which installedwind generation capacity and load grow 8 % and 3 % per year, respectively.By comparing results of the two simulations we saw that increasing share ofwind generation capacity increases RT price volatility in ERCOT market.

The statistical analysis of RT price and its potential drivers showed usthat occurrence of price spikes can be largely explained by explanatory vari-ables surplus capacity and DA price. We found that probability of price spikeis immensely higher when surplus capacity is below the limit of 7 GW thanwhen it is above the limit. On normal (non-spike) RT price level, forecasterror of surplus capacity, change speed of net-load, DA price, and previousbehaviour of RT price explain RT price movements.

We chose to build the forecasting method using bootstrap method since

99

CHAPTER 5. CONCLUSIONS 100

it makes no assumptions of specific distributions or the stochastic processesthat generate prices. There is also a strong theoretical ground for forecastingusing bootstrap method and it has been used in many applications. Wemade several modifications to account for impact of explanatory variables.New explanatory variables can be easily added to the model, if identified.We divided historical RT prices to several sampling groups. Conditions ofeach future time-step determine probabilities used in stochastic selection ofsampling groups.

5.1 Topics for future research

The model introduced here could be modified in several ways. Simulationof actual demand could be done taking into account different user groupsand modelling their demands separately, as ERCOT does. Autocorrelationof simulated actual wind generation and other explanatory variables couldbe made stronger. One possibility would be using block bootstrap method,instead of Markov chain and simple bootstrap of actual value as a percentage,in a similar way that we simulate RT price within spike periods.

Impact of solar generation could be added to the forecasting model asone explanatory variable that varies through the day stochastically, takinginto account the impact of season and hour of day. Also, differences betweennode and hub prices could be studied thoroughly to see impact of congestionon electricity price.

Accuracy of forecast RT price conditional on explanatory variable valuescould be improved by increasing the number of different dynamic time stepcategories by adding division points to explanatory variables. However, thereis limit to accuracy improvement using this method, since sampling groups si-multaneously decrease in size, reducing important randomness in simulation.Highest accuracy could be achieved by building a fundamental forecastingmodel based on explanatory variables predicted accurately, but it would bevery difficult to construct. RT price is much more volatile than DA priceand often exceeds significantly marginal generation costs of power plants,that form the basis for DA market simulations. Also, predicting explanatoryvariables precisely is impossible task.

Methodology developed in this work could also be applied to an othermarket with some modifications. It would only make sense in a marketwhere functioning RT market exists, of course. It is likely that price driverswould be different in some other market, and same parameter values as herecould certainly not be used. Steps needed to create a stochastic RT priceforecasting method are:

CHAPTER 5. CONCLUSIONS 101

1. Interview experts to understand RT price creation in the chosen market.Hypothesize how RT price could be modelled and what are the mostimportant drivers of RT price.

2. Conduct statistical analysis to decide whether RT price or spread be-tween RT and DA prices has to be simulated. Possibly both are needed,but in different situations depending on DA price as we did in this study.Decision depends on which variable has more uniform distribution.

3. Conduct statistical analysis (scatter plots, empirical autocorrelationfunction, empirical probabilities of price groups conditional on valuesof explanatory variables) to identify price drivers

4. Conduct statistical analysis (scatter plots, conditional histograms ofRT price) to select class limits for identified price drivers and neededlags of dependent variable as explanatory variable.

5. Build simulation methodology based on historical and future time seriesof price drivers in the same way as we did in this study.

6. Validate model functioning by comparing simulated and historical RTprice time series, when historical inputs are used for simulation.

7. Simulate future inputs using DA market modelling software and stochas-tic modelling.

8. Forecast future RT price using possibly more than one scenario of de-velopment of explanatory variables in future. Compare results to drawconclusions of future RT price volatility in the chosen market.

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Chapter 6

Dynamic time step categories

Table 6.1: Values of explanatory variables and category of previoustime step (delta class, price class or spike) of each dynamic timestep category C ∈ 0, 1, 2, ..., 300. C = 300 corresponds to pricespike.

C Xspike(t) ∆C(t− 1) RTC(t− 1) Xspike(t− 1) DA FCFE NLCA0 0 0 0 DA < 25 < −1800 < 2001 0 0 0 DA < 25 < −1800 ≥ 2002 0 0 0 DA < 25 ≥ −1800 < 2003 0 0 0 DA < 25 ≥ −1800 ≥ 2004 0 0 0 25 ≤ DA < 40 < −1800 < 2005 0 0 0 25 ≤ DA < 40 < −1800 ≥ 2006 0 0 0 25 ≤ DA < 40 ≥ −1800 < 2007 0 0 0 25 ≤ DA < 40 ≥ −1800 ≥ 2008 0 0 0 40 ≤ DA < 300 < −1800 < 2009 0 0 0 40 ≤ DA < 300 < −1800 ≥ 20010 0 0 0 40 ≤ DA < 300 ≥ −1800 < 20011 0 0 0 40 ≤ DA < 300 ≥ −1800 ≥ 20012 0 1 0 DA < 25 < −1800 < 20013 0 1 0 DA < 25 < −1800 ≥ 20014 0 1 0 DA < 25 ≥ −1800 < 20015 0 1 0 DA < 25 ≥ −1800 ≥ 20016 0 1 0 25 ≤ DA < 40 < −1800 < 20017 0 1 0 25 ≤ DA < 40 < −1800 ≥ 20018 0 1 0 25 ≤ DA < 40 ≥ −1800 < 20019 0 1 0 25 ≤ DA < 40 ≥ −1800 ≥ 20020 0 1 0 40 ≤ DA < 300 < −1800 < 20021 0 1 0 40 ≤ DA < 300 < −1800 ≥ 20022 0 1 0 40 ≤ DA < 300 ≥ −1800 < 20023 0 1 0 40 ≤ DA < 300 ≥ −1800 ≥ 20024 0 2 0 DA < 25 < −1800 < 20025 0 2 0 DA < 25 < −1800 ≥ 200

Continued on next page

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CHAPTER 6. DYNAMIC TIME STEP CATEGORIES 106

Table 6.1 – continued from previous pageC Xspike(t) ∆C(t− 1) RTC(t− 1) Xspike(t− 1) DA FCFE NLCA26 0 2 0 DA < 25 ≥ −1800 < 20027 0 2 0 DA < 25 ≥ −1800 ≥ 20028 0 2 0 25 ≤ DA < 40 < −1800 < 20029 0 2 0 25 ≤ DA < 40 < −1800 ≥ 20030 0 2 0 25 ≤ DA < 40 ≥ −1800 < 20031 0 2 0 25 ≤ DA < 40 ≥ −1800 ≥ 20032 0 2 0 40 ≤ DA < 300 < −1800 < 20033 0 2 0 40 ≤ DA < 300 < −1800 ≥ 20034 0 2 0 40 ≤ DA < 300 ≥ −1800 < 20035 0 2 0 40 ≤ DA < 300 ≥ −1800 ≥ 20036 0 3 0 DA < 25 < −1800 < 20037 0 3 0 DA < 25 < −1800 ≥ 20038 0 3 0 DA < 25 ≥ −1800 < 20039 0 3 0 DA < 25 ≥ −1800 ≥ 20040 0 3 0 25 ≤ DA < 40 < −1800 < 20041 0 3 0 25 ≤ DA < 40 < −1800 ≥ 20042 0 3 0 25 ≤ DA < 40 ≥ −1800 < 20043 0 3 0 25 ≤ DA < 40 ≥ −1800 ≥ 20044 0 3 0 40 ≤ DA < 300 < −1800 < 20045 0 3 0 40 ≤ DA < 300 < −1800 ≥ 20046 0 3 0 40 ≤ DA < 300 ≥ −1800 < 20047 0 3 0 40 ≤ DA < 300 ≥ −1800 ≥ 20048 0 4 0 DA < 25 < −1800 < 20049 0 4 0 DA < 25 < −1800 ≥ 20050 0 4 0 DA < 25 ≥ −1800 < 20051 0 4 0 DA < 25 ≥ −1800 ≥ 20052 0 4 0 25 ≤ DA < 40 < −1800 < 20053 0 4 0 25 ≤ DA < 40 < −1800 ≥ 20054 0 4 0 25 ≤ DA < 40 ≥ −1800 < 20055 0 4 0 25 ≤ DA < 40 ≥ −1800 ≥ 20056 0 4 0 40 ≤ DA < 300 < −1800 < 20057 0 4 0 40 ≤ DA < 300 < −1800 ≥ 20058 0 4 0 40 ≤ DA < 300 ≥ −1800 < 20059 0 4 0 40 ≤ DA < 300 ≥ −1800 ≥ 20060 0 5 0 DA < 25 < −1800 < 20061 0 5 0 DA < 25 < −1800 ≥ 20062 0 5 0 DA < 25 ≥ −1800 < 20063 0 5 0 DA < 25 ≥ −1800 ≥ 20064 0 5 0 25 ≤ DA < 40 < −1800 < 20065 0 5 0 25 ≤ DA < 40 < −1800 ≥ 20066 0 5 0 25 ≤ DA < 40 ≥ −1800 < 20067 0 5 0 25 ≤ DA < 40 ≥ −1800 ≥ 20068 0 5 0 40 ≤ DA < 300 < −1800 < 20069 0 5 0 40 ≤ DA < 300 < −1800 ≥ 20070 0 5 0 40 ≤ DA < 300 ≥ −1800 < 20071 0 5 0 40 ≤ DA < 300 ≥ −1800 ≥ 200



Table 6.1 – continued from previous pageC Xspike(t) ∆C(t− 1) RTC(t− 1) Xspike(t− 1) DA FCFE NLCA72 0 6 0 DA < 25 < −1800 < 20073 0 6 0 DA < 25 < −1800 ≥ 20074 0 6 0 DA < 25 ≥ −1800 < 20075 0 6 0 DA < 25 ≥ −1800 ≥ 20076 0 6 0 25 ≤ DA < 40 < −1800 < 20077 0 6 0 25 ≤ DA < 40 < −1800 ≥ 20078 0 6 0 25 ≤ DA < 40 ≥ −1800 < 20079 0 6 0 25 ≤ DA < 40 ≥ −1800 ≥ 20080 0 6 0 40 ≤ DA < 300 < −1800 < 20081 0 6 0 40 ≤ DA < 300 < −1800 ≥ 20082 0 6 0 40 ≤ DA < 300 ≥ −1800 < 20083 0 6 0 40 ≤ DA < 300 ≥ −1800 ≥ 20084 0 7 0 DA < 25 < −1800 < 20085 0 7 0 DA < 25 < −1800 ≥ 20086 0 7 0 DA < 25 ≥ −1800 < 20087 0 7 0 DA < 25 ≥ −1800 ≥ 20088 0 7 0 25 ≤ DA < 40 < −1800 < 20089 0 7 0 25 ≤ DA < 40 < −1800 ≥ 20090 0 7 0 25 ≤ DA < 40 ≥ −1800 < 20091 0 7 0 25 ≤ DA < 40 ≥ −1800 ≥ 20092 0 7 0 40 ≤ DA < 300 < −1800 < 20093 0 7 0 40 ≤ DA < 300 < −1800 ≥ 20094 0 7 0 40 ≤ DA < 300 ≥ −1800 < 20095 0 7 0 40 ≤ DA < 300 ≥ −1800 ≥ 20096 0 8 0 DA < 25 < −1800 < 20097 0 8 0 DA < 25 < −1800 ≥ 20098 0 8 0 DA < 25 ≥ −1800 < 20099 0 8 0 DA < 25 ≥ −1800 ≥ 200100 0 8 0 25 ≤ DA < 40 < −1800 < 200101 0 8 0 25 ≤ DA < 40 < −1800 ≥ 200102 0 8 0 25 ≤ DA < 40 ≥ −1800 < 200103 0 8 0 25 ≤ DA < 40 ≥ −1800 ≥ 200104 0 8 0 40 ≤ DA < 300 < −1800 < 200105 0 8 0 40 ≤ DA < 300 < −1800 ≥ 200106 0 8 0 40 ≤ DA < 300 ≥ −1800 < 200107 0 8 0 40 ≤ DA < 300 ≥ −1800 ≥ 200108 0 9 0 DA < 25 < −1800 < 200109 0 9 0 DA < 25 < −1800 ≥ 200110 0 9 0 DA < 25 ≥ −1800 < 200111 0 9 0 DA < 25 ≥ −1800 ≥ 200112 0 9 0 25 ≤ DA < 40 < −1800 < 200113 0 9 0 25 ≤ DA < 40 < −1800 ≥ 200114 0 9 0 25 ≤ DA < 40 ≥ −1800 < 200115 0 9 0 25 ≤ DA < 40 ≥ −1800 ≥ 200116 0 9 0 40 ≤ DA < 300 < −1800 < 200117 0 9 0 40 ≤ DA < 300 < −1800 ≥ 200



Table 6.1 – continued from previous pageC Xspike(t) ∆C(t− 1) RTC(t− 1) Xspike(t− 1) DA FCFE NLCA

118 0 9 0 40 ≤ DA < 300 ≥ −1800 < 200119 0 9 0 40 ≤ DA < 300 ≥ −1800 ≥ 200120 0 10 0 DA < 25 < −1800 < 200121 0 10 0 DA < 25 < −1800 ≥ 200122 0 10 0 DA < 25 ≥ −1800 < 200123 0 10 0 DA < 25 ≥ −1800 ≥ 200124 0 10 0 25 ≤ DA < 40 < −1800 < 200125 0 10 0 25 ≤ DA < 40 < −1800 ≥ 200126 0 10 0 25 ≤ DA < 40 ≥ −1800 < 200127 0 10 0 25 ≤ DA < 40 ≥ −1800 ≥ 200128 0 10 0 40 ≤ DA < 300 < −1800 < 200129 0 10 0 40 ≤ DA < 300 < −1800 ≥ 200130 0 10 0 40 ≤ DA < 300 ≥ −1800 < 200131 0 10 0 40 ≤ DA < 300 ≥ −1800 ≥ 200132 0 11 0 DA < 25 < −1800 < 200133 0 11 0 DA < 25 < −1800 ≥ 200134 0 11 0 DA < 25 ≥ −1800 < 200135 0 11 0 DA < 25 ≥ −1800 ≥ 200136 0 11 0 25 ≤ DA < 40 < −1800 < 200137 0 11 0 25 ≤ DA < 40 < −1800 ≥ 200138 0 11 0 25 ≤ DA < 40 ≥ −1800 < 200139 0 11 0 25 ≤ DA < 40 ≥ −1800 ≥ 200140 0 11 0 40 ≤ DA < 300 < −1800 < 200141 0 11 0 40 ≤ DA < 300 < −1800 ≥ 200142 0 11 0 40 ≤ DA < 300 ≥ −1800 < 200143 0 11 0 40 ≤ DA < 300 ≥ −1800 ≥ 200144 0 0 0 DA < 25 < −1800 < 200145 0 0 0 DA < 25 < −1800 ≥ 200146 0 0 0 DA < 25 ≥ −1800 < 200147 0 0 0 DA < 25 ≥ −1800 ≥ 200148 0 0 0 25 ≤ DA < 40 < −1800 < 200149 0 0 0 25 ≤ DA < 40 < −1800 ≥ 200150 0 0 0 25 ≤ DA < 40 ≥ −1800 < 200151 0 0 0 25 ≤ DA < 40 ≥ −1800 ≥ 200152 0 0 0 40 ≤ DA < 300 < −1800 < 200153 0 0 0 40 ≤ DA < 300 < −1800 ≥ 200154 0 0 0 40 ≤ DA < 300 ≥ −1800 < 200155 0 0 0 40 ≤ DA < 300 ≥ −1800 ≥ 200156 0 1 0 DA < 25 < −1800 < 200157 0 1 0 DA < 25 < −1800 ≥ 200158 0 1 0 DA < 25 ≥ −1800 < 200159 0 1 0 DA < 25 ≥ −1800 ≥ 200160 0 1 0 25 ≤ DA < 40 < −1800 < 200161 0 1 0 25 ≤ DA < 40 < −1800 ≥ 200162 0 1 0 25 ≤ DA < 40 ≥ −1800 < 200163 0 1 0 25 ≤ DA < 40 ≥ −1800 ≥ 200




164 0 1 0 40 ≤ DA < 300 < −1800 < 200165 0 1 0 40 ≤ DA < 300 < −1800 ≥ 200166 0 1 0 40 ≤ DA < 300 ≥ −1800 < 200167 0 1 0 40 ≤ DA < 300 ≥ −1800 ≥ 200168 0 2 0 DA < 25 < −1800 < 200169 0 2 0 DA < 25 < −1800 ≥ 200170 0 2 0 DA < 25 ≥ −1800 < 200171 0 2 0 DA < 25 ≥ −1800 ≥ 200172 0 2 0 25 ≤ DA < 40 < −1800 < 200173 0 2 0 25 ≤ DA < 40 < −1800 ≥ 200174 0 2 0 25 ≤ DA < 40 ≥ −1800 < 200175 0 2 0 25 ≤ DA < 40 ≥ −1800 ≥ 200176 0 2 0 40 ≤ DA < 300 < −1800 < 200177 0 2 0 40 ≤ DA < 300 < −1800 ≥ 200178 0 2 0 40 ≤ DA < 300 ≥ −1800 < 200179 0 2 0 40 ≤ DA < 300 ≥ −1800 ≥ 200180 0 3 0 DA < 25 < −1800 < 200181 0 3 0 DA < 25 < −1800 ≥ 200182 0 3 0 DA < 25 ≥ −1800 < 200183 0 3 0 DA < 25 ≥ −1800 ≥ 200184 0 3 0 25 ≤ DA < 40 < −1800 < 200185 0 3 0 25 ≤ DA < 40 < −1800 ≥ 200186 0 3 0 25 ≤ DA < 40 ≥ −1800 < 200187 0 3 0 25 ≤ DA < 40 ≥ −1800 ≥ 200188 0 3 0 40 ≤ DA < 300 < −1800 < 200189 0 3 0 40 ≤ DA < 300 < −1800 ≥ 200190 0 3 0 40 ≤ DA < 300 ≥ −1800 < 200191 0 3 0 40 ≤ DA < 300 ≥ −1800 ≥ 200192 0 4 0 DA < 25 < −1800 < 200193 0 4 0 DA < 25 < −1800 ≥ 200194 0 4 0 DA < 25 ≥ −1800 < 200195 0 4 0 DA < 25 ≥ −1800 ≥ 200196 0 4 0 25 ≤ DA < 40 < −1800 < 200197 0 4 0 25 ≤ DA < 40 < −1800 ≥ 200198 0 4 0 25 ≤ DA < 40 ≥ −1800 < 200199 0 4 0 25 ≤ DA < 40 ≥ −1800 ≥ 200200 0 4 0 40 ≤ DA < 300 < −1800 < 200201 0 4 0 40 ≤ DA < 300 < −1800 ≥ 200202 0 4 0 40 ≤ DA < 300 ≥ −1800 < 200203 0 4 0 40 ≤ DA < 300 ≥ −1800 ≥ 200204 0 5 0 DA < 25 < −1800 < 200205 0 5 0 DA < 25 < −1800 ≥ 200206 0 5 0 DA < 25 ≥ −1800 < 200207 0 5 0 DA < 25 ≥ −1800 ≥ 200208 0 5 0 25 ≤ DA < 40 < −1800 < 200209 0 5 0 25 ≤ DA < 40 < −1800 ≥ 200




210 0 5 0 25 ≤ DA < 40 ≥ −1800 < 200211 0 5 0 25 ≤ DA < 40 ≥ −1800 ≥ 200212 0 5 0 40 ≤ DA < 300 < −1800 < 200213 0 5 0 40 ≤ DA < 300 < −1800 ≥ 200214 0 5 0 40 ≤ DA < 300 ≥ −1800 < 200215 0 5 0 40 ≤ DA < 300 ≥ −1800 ≥ 200216 0 6 0 DA < 25 < −1800 < 200217 0 6 0 DA < 25 < −1800 ≥ 200218 0 6 0 DA < 25 ≥ −1800 < 200219 0 6 0 DA < 25 ≥ −1800 ≥ 200220 0 6 0 25 ≤ DA < 40 < −1800 < 200221 0 6 0 25 ≤ DA < 40 < −1800 ≥ 200222 0 6 0 25 ≤ DA < 40 ≥ −1800 < 200223 0 6 0 25 ≤ DA < 40 ≥ −1800 ≥ 200224 0 6 0 40 ≤ DA < 300 < −1800 < 200225 0 6 0 40 ≤ DA < 300 < −1800 ≥ 200226 0 6 0 40 ≤ DA < 300 ≥ −1800 < 200227 0 6 0 40 ≤ DA < 300 ≥ −1800 ≥ 200228 0 7 0 DA < 25 < −1800 < 200229 0 7 0 DA < 25 < −1800 ≥ 200230 0 7 0 DA < 25 ≥ −1800 < 200231 0 7 0 DA < 25 ≥ −1800 ≥ 200232 0 7 0 25 ≤ DA < 40 < −1800 < 200233 0 7 0 25 ≤ DA < 40 < −1800 ≥ 200234 0 7 0 25 ≤ DA < 40 ≥ −1800 < 200235 0 7 0 25 ≤ DA < 40 ≥ −1800 ≥ 200236 0 7 0 40 ≤ DA < 300 < −1800 < 200237 0 7 0 40 ≤ DA < 300 < −1800 ≥ 200238 0 7 0 40 ≤ DA < 300 ≥ −1800 < 200239 0 7 0 40 ≤ DA < 300 ≥ −1800 ≥ 200240 0 8 0 DA < 25 < −1800 < 200241 0 8 0 DA < 25 < −1800 ≥ 200242 0 8 0 DA < 25 ≥ −1800 < 200243 0 8 0 DA < 25 ≥ −1800 ≥ 200244 0 8 0 25 ≤ DA < 40 < −1800 < 200245 0 8 0 25 ≤ DA < 40 < −1800 ≥ 200246 0 8 0 25 ≤ DA < 40 ≥ −1800 < 200247 0 8 0 25 ≤ DA < 40 ≥ −1800 ≥ 200248 0 8 0 40 ≤ DA < 300 < −1800 < 200249 0 8 0 40 ≤ DA < 300 < −1800 ≥ 200250 0 8 0 40 ≤ DA < 300 ≥ −1800 < 200251 0 8 0 40 ≤ DA < 300 ≥ −1800 ≥ 200252 0 9 0 DA < 25 < −1800 < 200253 0 9 0 DA < 25 < −1800 ≥ 200254 0 9 0 DA < 25 ≥ −1800 < 200255 0 9 0 DA < 25 ≥ −1800 ≥ 200




256 0 9 0 25 ≤ DA < 40 < −1800 < 200257 0 9 0 25 ≤ DA < 40 < −1800 ≥ 200258 0 9 0 25 ≤ DA < 40 ≥ −1800 < 200259 0 9 0 25 ≤ DA < 40 ≥ −1800 ≥ 200260 0 9 0 40 ≤ DA < 300 < −1800 < 200261 0 9 0 40 ≤ DA < 300 < −1800 ≥ 200262 0 9 0 40 ≤ DA < 300 ≥ −1800 < 200263 0 9 0 40 ≤ DA < 300 ≥ −1800 ≥ 200264 0 10 0 DA < 25 < −1800 < 200265 0 10 0 DA < 25 < −1800 ≥ 200266 0 10 0 DA < 25 ≥ −1800 < 200267 0 10 0 DA < 25 ≥ −1800 ≥ 200268 0 10 0 25 ≤ DA < 40 < −1800 < 200269 0 10 0 25 ≤ DA < 40 < −1800 ≥ 200270 0 10 0 25 ≤ DA < 40 ≥ −1800 < 200271 0 10 0 25 ≤ DA < 40 ≥ −1800 ≥ 200272 0 10 0 40 ≤ DA < 300 < −1800 < 200273 0 10 0 40 ≤ DA < 300 < −1800 ≥ 200274 0 10 0 40 ≤ DA < 300 ≥ −1800 < 200275 0 10 0 40 ≤ DA < 300 ≥ −1800 ≥ 200276 0 11 0 DA < 25 < −1800 < 200277 0 11 0 DA < 25 < −1800 ≥ 200278 0 11 0 DA < 25 ≥ −1800 < 200279 0 11 0 DA < 25 ≥ −1800 ≥ 200280 0 11 0 25 ≤ DA < 40 < −1800 < 200281 0 11 0 25 ≤ DA < 40 < −1800 ≥ 200282 0 11 0 25 ≤ DA < 40 ≥ −1800 < 200283 0 11 0 25 ≤ DA < 40 ≥ −1800 ≥ 200284 0 11 0 40 ≤ DA < 300 < −1800 < 200285 0 11 0 40 ≤ DA < 300 < −1800 ≥ 200286 0 11 0 40 ≤ DA < 300 ≥ −1800 < 200287 0 11 0 40 ≤ DA < 300 ≥ −1800 ≥ 200288 0 1 DA < 25 < −1800 < 200289 0 1 DA < 25 < −1800 ≥ 200290 0 1 DA < 25 ≥ −1800 < 200291 0 1 DA < 25 ≥ −1800 ≥ 200292 0 1 25 ≤ DA < 40 < −1800 < 200293 0 1 25 ≤ DA < 40 < −1800 ≥ 200294 0 1 25 ≤ DA < 40 ≥ −1800 < 200295 0 1 25 ≤ DA < 40 ≥ −1800 ≥ 200296 0 1 40 ≤ DA < 300 < −1800 < 200297 0 1 40 ≤ DA < 300 < −1800 ≥ 200298 0 1 40 ≤ DA < 300 ≥ −1800 < 200299 0 1 40 ≤ DA < 300 ≥ −1800 ≥ 200300 1

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